In this chapter you
will learn:
• about different units for
measuring angles, and
how measuring angles
is related to distance
travelled around a circle
9
• the definitions of sine,
cosine and tangent
functions, their basic
properties and their
graphs
• how to calculate
certain special values
of trigonometric
functions
• how to apply your
knowledge of
transformations of
graphs to sketch
more complicated
trigonometric functions
• how to use
trigonometric functions
to model periodic
phenomena
• about inverses of
trigonometric functions.
Circular
measure and
trigonometric
functions
Introductory problem
The original Ferris Wheel was constructed in 1893 in
Chicago. It was just over 80 m tall and could complete one
full revolution in 9 minutes. During each revolution, how
long did the passengers spend more than 5 m above ground?
Measuring angles is related to measuring lengths around
the perimeter of the circle. This observation leads to the
introduction of a new unit for measuring angles, the radian,
which will be a very useful unit of measurement in advanced
mathematics.
Motion in a circle is just one example of periodic motion, which
repeats after a fixed time interval. Other examples include
oscillation of a particle attached to the end of a spring or
vibration of a guitar string. There are also periodic phenomena
where a pattern is repeated in space rather than in time; for
example, the shape of a water wave. All these can be modelled
using trigonometric functions.
9A Radian measure
An angle measures the amount of rotation between two straight
lines. You are already familiar with measuring angles in degrees,
where a full turn measures 360o and that there are two directions
(or senses) of rotation; clockwise and anti-clockwise. It is
conventional in mathematics to measure angles anti-clockwise.
232 Topic 3: Circular functions and trigonometry
© Cambridge University Press 2012
Not for printing, sharing or distribution.
B
In the first diagram alongside, the line OA rotates 60o anticlockwise into position OB. Therefore ΑÔΒ = 60°.
In the second diagram the line OA rotates 150o clockwise into
position OC. We measure clockwise rotations with negative
angles. Therefore AÔC = −150°. Note that OA can also
move to position OC by rotating 210o anti-clockwise, so also
AÔC = 210°, as shown in the third diagram. In this book, and
in exam questions, it will always be made clear which of the two
angles is required.
The measure of 360o for a full turn may seem arbitrary and
there are other ways of measuring sizes of angles. In advanced
mathematics, the most useful unit for measuring angles is the
radian. This measure relates the size of the angle to the distance
moved by a point around a circle.
Consider a circle with centre O and radius 1 (this is the unit
circle), and two points, A and B, on its circumference. As the
line OA rotates into position OB, point A moves the distance
equal to the length of the arc AB. The measure of the angle
AOB in radians is equal to this arc length.
If point A makes a full rotation around the circle, it will cover a
distance equal to the length of the circumference of the circle.
As the radius of the circle is 1, the length of the circumference
is 2π. Hence a full turn measures 2π radians. We can deduce the
sizes of other common angles in radians; for example, a right
π
angle is one quarter of a full turn, so it measures 2π 4 =
2
radians.
Common angles, measured in radians, are often expressed as
fractions of π but we can also use decimal approximations. Thus
a right angle measures approximately 1.57 radians.
The fact that a full turn measures 2π radians can be used to
convert any angle measurement from degrees to radians,
and vice versa.
© Cambridge University Press 2012
O
60◦
A
O
◦
150
A
C
◦
210
O
A
C
B
O
You may wonder
why there are 360
degrees in a full
turn. It came from
ancient astrologers who
noticed that the stars
appeared to rotate in the
sky, returning to their original
position after around 360
days. One day’s rotation
corresponds to one degree.
We now know that there are
365.24 days in a year, so
these ancient astrologers
were quite accurate – you
might want to think how you
could measure how many
days there are in a year.
9 Circular measure and trigonometric functions
Not for printing, sharing or distribution.
A
233
Worked example 9.1
(a) Convert 75° to radians.
(b) Convert 2.5 radians to degrees.
What fraction of a full turn is 75°?
(a)
5
5π
× 2π
2 =
24
12
Calculate the same fraction of 2π
5π
radians
12
75° = 1.31
(3SF)
∴ 75° =
This is the exact answer. We can
also find the decimal equivalent,
to 3 significant figures.
What fraction of a full turn is
2.5 radians?
75
5
=
360 24
(b)
Calculate the same fraction of 360°
25
( 0.3979)
2π
25
× 360
2π
143.24 …
1143
43° (3SF)
2.5
KEY POINT 9.1
Full turn = 360° = 2π radians
There are many
different measures of
angle. One historical
attempt was gradians,
which split a right angle into
100 units. Does this mean that
the facts you have learnt about
angles – ideas such as 180°
in a triangle – are purely
consequences of definitions
and have no link to truth?
Half turn = 180° = π radians
To convert from degrees to radians, divide by 180 and
multiply by π.
To convert from radians to degrees, divide by π and
multiply by 180.
In our definition of the radian measure we have used the unit
circle. However, we can think about a point moving around a
circle of any radius.
B
l
r
θ
O
r
A
As the line OA rotates into position OB, point A covers the
distance equal to the length of the arc AB. The ratio of this
l
arc length l to the circumference of the circle is
.
2πr
The measure of AÔB in radians is in the same ratio to the full
θ
l
turn, so
.
=
2π 2πr
l
It follows that the measure of the angle in radians is θ = ;
r
the measure of the angle is the ratio of the length of the arc
234 Topic 3: Circular functions and trigonometry
© Cambridge University Press 2012
Not for printing, sharing or distribution.
to the radius of the circle. In particular, the angle of 1 radian
corresponds to an arc whose length is equal to the radius of the
circle.
When the radius of the circle is 1, the size of the angle is
numerically equal to the length of the arc. However, as the size
of the angle is a ratio of two lengths, it has no units. We say that
the radian is a dimensionless unit. For this reason, when writing
the size of an angle in radians, we will simply write, for example,
θ = 1.31.
All this may sound a little complicated and you may wonder
why we cannot just use degrees to measure angles. You will see
in chapter 11 that the formulae for calculating lengths and areas
for parts of circles are much simpler when radians are used, but
an even stronger case for using radians will be presented when
you study calculus.
One advantage of thinking of angles as measuring the amount
of rotation around the unit circle is that we can show an angle of
any size by marking the corresponding point on the unit circle.
We have seen that the convention in mathematics is to measure
positive angles anti-clockwise. Another convention is that we
start measuring from a point which is on the positive x-axis. On
the diagram alongside, the starting point is labelled A, and the
point P corresponds to the angle of 60°. In other words, to get
from the starting point to point P, we need to rotate 60°, or onesixth of the full turn, around the circle.
We can also represent negative angles (by rotating clockwise)
and angles larger than 360° (by rotating through more than a
full turn). The second diagram shows point P representing the
angle of 450°, or one and a quarter turns.
We can apply this idea to representing numbers; instead of
representing numbers by points on a number line, we can
represent them by points on the unit circle. To do this, we
imagine wrapping the number line around a circle of radius 1,
starting by placing zero on the positive x-axis (on the diagram
on the next page, point S corresponds to number 0), and going
anti-clockwise. As the length of the circumference of the circle
is 2π, the numbers 2π, 4π, and so on are also represented by
point S. Numbers π, 3π, and so on are represented by point P.
Number 3, which is just less than π, is represented by point A as
shown in the diagram on the next page.
© Cambridge University Press 2012
t
exam hin
s write
Some book
d or
θ = 1.31 cra
ut the
b
,
θ = 1.31
use
t
o
IB® does n
.
n
this notatio
Radians will be
used
whenever
differentiating
and
integrating
trigonometric
functions (chapters
16 to 20).
P
◦
60
O
A
P
◦
A
450
There is a threedimensional analogue
of angle called solid
angle, the unit of which is
steradians. This unit
measures the fraction of the
surface area of a sphere
covered. There are many
aspects of trigonometry
which can be transferred to
these solid angles.
9 Circular measure and trigonometric functions
Not for printing, sharing or distribution.
235
Looking at the
number line in this
way should also
suggest why radians
are a more natural measure
of angle than degrees.
While the radian is rarely
used outside of the
mathematical community,
within mathematics it is the
unit of choice.
S
B
2
1
A
P
S
1
2
3
We can also represent negative numbers in this way, by
wrapping the negative part of the number line clockwise around
the circle. For example, number –3 is represented by point B
on the diagram alongside and number –7 by point C (7 is a bit
bigger than 2π, which means wrapping once and a bit around
the circle).
C
Worked example 9.2
(a) Mark on the unit circle the points corresponding to the following angles, measured in
degrees:
A: 135° B: 270°
C: –120°
D: 765°
(b) Mark on the unit circle the points corresponding to the following angles, measured in
radians:
π
5π
13π
A: π
B: −
C:
D:
2
2
3
135 = 90 + 45, so point A
represents quarter of a turn plus another
eighth of a turn
(a)
A
D
270 = 3 × 90, so point B represents
rotation through three right angles
120 = 360 ÷ 3, so point C represents
a third of the full turn, but clockwise
(because of the minus sign)
C
B
765 = 2 × 360 + 45, so point D
represents two full turns plus one half of
a right angle
236 Topic 3: Circular functions and trigonometry
© Cambridge University Press 2012
Not for printing, sharing or distribution.
continued . . .
π radians is one half of a full turn
so point A represents half a turn
(b)
C
D
π
is one quarter of a full turn, so
2
point B represents a quarter of
a turn in the clockwise direction
(because of the minus sign)
A
5π
π
= 2π
2π + , so point C represents
2
2
a full turn followed by another
quarter of a turn
B
13π
π
= 4π
4π + , so point D represents
3
3
two full turns followed by another
6th of a turn
A
It is useful to describe where points are on the unit circle
using quadrants. A quadrant is one quarter of the circle, and
conventionally quadrants are labelled going anti-clockwise. So
in the example (a) above, point A is in the second quadrant,
point C in the third quadrant and point D in the first quadrant.
D
C
B
Exercise 9A
1. Draw a unit circle for each part and mark the points
corresponding to the following angles.
(a) (i) 60°
(ii) 150°
(b) (i) −120°
(ii) −90°
(c) (i) 495°
(ii) 390°
2. Draw a unit circle for each part and mark the points
corresponding to the following angles.
π
π
(a) (i)
(ii)
4
3
4π
3π
(ii)
(b) (i)
3
4
© Cambridge University Press 2012
9 Circular measure and trigonometric functions
Not for printing, sharing or distribution.
237
π
3
(d) (i) −2π
(c) (i) −
π
6
(ii) −4π
(ii) −
3. Express the following angles in radians, giving your answer in
terms of π.
(a) (i) 135°
(ii)
(b) (i) 90°
(ii) 270°
(c) (i) 120°
(ii) 150°
(d) (i) 50°
(ii) 80°
45°
4. Express the following angles in radians, correct to 3 decimal places.
(a) (i) 320°
(ii) 20°
(b) (i) 270°
(ii) 90°
(c) (i) 65°
(ii) 145°
(d) (i) 100°
(ii) 83°
5. Express the following angles in degrees.
π
π
(a) (i)
(ii)
3
4
5π
2π
(b) (i)
(ii)
6
3
3π
5π
(c) (i)
(ii)
2
3
(d) (i) 1.22
(ii) 4.63
6. (a) (i) Express 42o in radians to 3 decimal places.
(ii) Express 168o in radians correct to 3 decimal places.
(b) (i) Express 202.5o in radians in terms of π.
(ii) Express 67.5o in radians in terms of π.
5π
(c) (i) Express
radians in degrees.
12
11π
radians in degrees.
(ii) Express
20
(d) (i) Express 1.62 radians in degrees to one decimal place.
P
O
θ
(ii) Express 2.73 radians in degrees to one decimal place.
7. The diagram shows point P on the unit circle corresponding to
angle θ (measured in degrees).
Copy the diagram and mark the points corresponding to the
following angles.
(a) (i) 180° − θ
(ii) 180° + θ
(b) (i) θ + 180°
(ii) θ + 90°
238 Topic 3: Circular functions and trigonometry
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(c) (i) 90° – θ
(ii) 270° − θ
(d) (i) θ − 360°
(ii) θ + 360°
8. The diagram shows point Q on the unit circle corresponding to
angle θ (measured in radians).
Copy the diagram and mark the points corresponding to the
following angles.
(a) (i) 2π − θ
(ii) π − θ
(b) (i) θ + π
π
+θ
(c) (i)
2
(d) (i) θ − 2π
(ii) −π − θ
π
(ii)
−θ
2
(ii) θ + 2π
Q
O
θ
9B Definitions and graphs of sine and
cosine functions
We will now define trigonometric functions. Starting with a real
number α, first mark a point A on the unit circle representing
the number α. The sine and cosine of this number is defined in
terms of distance to the axes.
KEY POINT 9.2
The sine of the number α, written sin α, is the distance of
the point A from the horizontal axis.
The cosine of the number α, written cos α, is the distance
of the point A from the vertical axis.
cos α
sin α
© Cambridge University Press 2012
A
α
You have previously
seen sine and cosine
defined using rightangled triangles.
See Prior learning
Section U
on the CD-ROM
The definition given here
is consistent with this but it
allows us to define sine and
cosine for values beyond
90°. This leads to the
question about what makes
something a definition. Is
it the one that came first
historically, the one that is
more understandable or
the one that is easier to
generalise?
9 Circular measure and trigonometric functions
Not for printing, sharing or distribution.
239
Worked example 9.3
By marking the corresponding points on the unit circle, estimate the value of:
(a) sin1.2
(a)
(b) cos6
(c) cos2.4
π
≈ 1.6, so the point is in the
2
first quadrant, close to
the top
sin 1.2 ≈ 0.9
(b)
2π
6.2, so the point is just
below the horizontal axis
cos 6 0.95
(c)
π
≈ 1.6 and π ≈ 3.1 so the point
2
is around the middle of the
second quadrant
cos 2.4 ≈ −0.7
240 Topic 3: Circular functions and trigonometry
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Notice that in the last example cos 2.4 was a negative number.
This is because the point corresponding to number 2.4 is to the
left of the vertical axis, so we take the distance to be negative.
You may be wondering why we had to change the calculator into
radian mode when we are working with real numbers rather
than angles. Remember how we defined the radian measure: the
size of the angle in radians is equal to the distance travelled by
a point around the unit circle as it rotates through that angle.
So measuring an angle in radians is the same as measuring the
distance ‘wrapped’ around the circle, which corresponds to
real numbers.
However, we can also think of a point on the unit circle as
corresponding to an angle measured in degrees. For example,
point A on the diagram corresponds to the real number 1.2,
to an angle of 1.2 radians and to an angle of 69°. If you change
your calculator into degree mode and calculate sin 69°, you will
get the same answer as when you worked out sin 1.2 in radian
mode. Although in applications the sine and cosine functions
are frequently used in situations with angles, you should
be aware that they are functions which can operate on any
real number.
We can use the symmetry of the circle to see how the values of
sine and cosine functions of different numbers are related to
each other.
t
exam hin
d the
You can fin
ne and
values of si
tions
cosine func
using your
ake
calculator. M
ulator is
lc
sure your ca
ode.
in radian m
A
1.2
◦
69
t
exam hin
always
You should
unless
use radians
ld to use
explicitly to
degrees.
Worked example 9.4
Given that sinθ = 0.6, find the values of:
(a) sin(π − θ)
(b) sin(θ + π)
Mark the points corresponding to
θ and π − θ on the circle
(a)
θ
© Cambridge University Press 2012
θ
9 Circular measure and trigonometric functions
Not for printing, sharing or distribution.
241
continued . . .
Compare the distances from the
horizontal axis
Mark the points corresponding to
θ and π + θ on the circle
The points are the same distance from the
horizontal axis, so sin( θ ) = 0.6
(b)
θ
θ
Compare the distances from the
horizontal axis
The points are the same distance from the
horizontal axis, but the second one is below the axis,
so sin (θ π ) = −0 6
The above example illustrates some of the properties of
the sine function. Similar properties can be derived for
the cosine function. It may be useful to remember those
results, summarised below. They can all be derived using
circle diagrams.
KEY POINT 9.3
π −x
x or x + 2π
sin x
−x
242 Topic 3: Circular functions and trigonometry
) = sin(x
sin(
i ( + 2 π)
sin(π + x)) sin(
cos x
π +x
sin(
cos(π
cos(
) = − sin x
) = cos(x
cos( x + 2π)
) cos(
c (
) = − cos x
© Cambridge University Press 2012
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Worked example 9.5
Given that sin x = 0.4 , find the value of:
π⎞
⎛π
⎞
⎛
−x
(b) cos x +
(a) cos
⎝2
⎠
⎝
2⎠
x,
x
B
Label the points corresponding to
x
Y
π
π
− x and x + on the
2
2
circle
A
P
x
X
⎛π
⎞
− x is represented by point
⎝2
⎠
A, and AY = PX
π
(a) cos ⎛ − x⎞ = 0 4
⎝2
⎠
π⎞
⎛
x+
is represented by point B,
⎝
2⎠
and BY = PX, but B is to the left of
the vertical axis
(b)
π⎞
⎛
s x+
⎝
2⎠
0.4
The above example illustrates a relationship between sine and
cosine functions. It may be a good idea to remember, or know
how to derive, the following results:
KEY POINT 9.4
π
cos ⎛
⎝2
π
cos ⎛ +
⎝2
π
sin ⎛
⎝2
π
sin ⎛ +
⎝2
⎞
⎠
⎞
⎠
sin x
π
2
+x
π
2 −x
x
sin x
⎞
⎠
cos x
⎞
⎠
cos x
Similar results can be derived when θ represents an angle
measured in degrees.
© Cambridge University Press 2012
9 Circular measure and trigonometric functions
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243
Worked example 9.6
Given that θ is an angle measured in degrees such that cosθ = 0.8 find the value of
(a) cos(180° + θ )
(b) sin(90° − θ )
Mark the points corresponding to
angles θ, 180° + θ and 90° − θ
on the circle
(b) 90° + θ
θ
180° + θ (a)
The distance from the vertical axis
is 0.8, but the point is to the left
of the axis
(a) cos (180° + θ ) = −0.8
By reflection symmetry in the
diagonal line, the distance from the
horizontal axis is 0.8
(b) sin (90° + θ ) = 0.8
We have now defined the sine function for all real numbers, so
we can draw its graph. To do this, we go back to thinking about
the real number line wrapped around the unit circle. Each real
number corresponds to a point on the circle, and the value
of the sine function is the distance of the point from the
horizontal axis.
x
0
244 Topic 3: Circular functions and trigonometry
x
© Cambridge University Press 2012
Not for printing, sharing or distribution.
All of the properties of the sine function discussed above
can be seen from the graph. For example, increasing x by
2π corresponds to making a full turn around the circle and
returning to the same point. Therefore sin( x + 2π) sin x.
We say that the sine function is periodic with period 2π. By
considering points A and B on the graph below, we can see that
sin( −x )
sin x . We can also see that the minimum possible
value of sin x is –1 and the maximum value is 1. We say that the
sine function has amplitude 1.
You may wonder
how your calculator
finds the values of
sine and cosine
functions. This is explored
in Supplementary sheet 8
‘How does your calculator
work out sin and cos?’ on
the CD-ROM.
y
1
A
−2π
−π
C
π
B
2π
x
−1
KEY POINT 9.5
A function is periodic if it repeats regularly. The distance
between repeats is called the period.
The amplitude is half of the distance between the maximum
and minimum value of a function.
To draw the graph of the cosine function we look at the distance
of the point representing real number x from the vertical axis.
y
x
0
1
x
−2π
−π
0
π
2π
x
−1
We can again see many of the properties discussed above on this
graph; for example, cos( −x ) cos x and cos(π − x )
cos x .
The period and the amplitude are the same as for the sine
function. In fact, the graphs of the sine and the cosine functions
are related to each other: the graph of y = cos x is obtained
from the graph of y = sin x by translating it π units to the left.
2
This again corresponds to one of the properties listed above:
π
cos x sin
si ⎛ x + ⎞ .
⎝
2⎠
You can see from the graphs that sin x is an odd function
and cos x is an even function. Both graphs have translational
symmetry. We discussed these symmetries in Section 6G.
© Cambridge University Press 2012
9 Circular measure and trigonometric functions
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245
Trigonometric
functions can be used
to define polar
coordinates. This
alternative to the Cartesian
coordinate system makes it
easier to write equations of
certain graphs. Such
equations produce some
beautiful curves, such as the
cardioid and the polar rose.
The key properties of sine and cosine functions are summarised
below.
KEY POINT 9.6
The sine and cosine functions are periodic with period 2π:
sin x
s (
sin(
) = sin(x
sin(
i ( ± 4 π) = …
cos x
cos((
) = cos(x
cos(( ± 4 π) = …
The sine and cosine functions have amplitude 1:
−1 ≤ sin ≤ 1
d −11 ≤ cos x ≤ 1
The cosine graph is obtained by translating the sine graph
π
units to the left:
2
π
π
cos x sin
si ⎛ x + ⎞ and sin
i x cos ⎛ x − ⎞
⎝
⎠
⎝
2
2⎠
Exercise 9B
1. Write down the approximate values of sin x and cos x for the
number x corresponding to each of the points marked on the
diagram
(a)
(c)
(b)
2. Use the unit circle to find the value of:
π
(a) (i) sin
(ii) sin2π
2
(b) (i) cos0
(ii) cos( −π)
5π
π
(c) (i) sin ⎛ − ⎞
(ii) cos
⎝ 2⎠
2
3. Use the unit circle to find the value of:
(a) (i) cos90°
(ii) cos180°
(b) (i) sin270°
(ii) sin90°
(c) (i) sin 270°
(ii) cos 450°
246 Topic 3: Circular functions and trigonometry
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4. Given that cos
4π
5
9π
(c) cos
5
π
5
0.809 find the value of:
21π
5
6π
(d) cos
5
(a) cos
(b) cos
5. Given that sin
−2π ⎞
(a) sin ⎛
⎝ 3 ⎠
10π
(c) sin
3
2π
3
0.866 find the value of:
4π
3
π
(d) sin
3
(b) sin
6. Given that cos 40° 0.766 find the value of:
(a) cos 400°
(b) cos320°
(c) cos(−220°)
(d) cos140°
7. Given that sin 130° 0.766 find the value of:
(a) sin 490°
(b) sin50°
(c) sin(−130°)
(d) sin230°
8. Sketch the graph of y
sin x for:
(a) (i) 0° ≤ x ≤ 180°
(b) (i) −
(ii) 90° ≤ x ≤ 360°
π
π
≤x≤
2
2
9. Sketch the graph of y
(ii) − π ≤
≤ 2π
cos x for:
(a) (i) −180° ≤ x ≤ 180°
π
3π
≤x≤
(b) (i)
2
2
(ii) 0° ≤ x ≤ 270°
(ii) − π ≤
≤ 2π
10. (a) On the unit circle, mark the points representing
(b) Given that sin
(i) cos
π
3
π
6
π π
2π
, and .
6 3
3
0.5 , find the value of:
(ii) cos
2π
3
11. Use your calculator to evaluate the following, giving your
answers to 3 significant figures.
(a) (i) cos1.25
(b) (i) cos(
.72)
(ii) sin0.68
(ii) sin( −2.. )
© Cambridge University Press 2012
9 Circular measure and trigonometric functions
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247
12. Use your calculator to evaluate the following, giving your
answers to 3 significant figures.
(a) (i) sin 42°
(ii) cos168°
(b) (i) sin( −50°)
(ii) cos( −227°)
13. Evaluate cos(π + ) cos
c ( π − ).
[4 marks]
[6 marks]
14. Simplify the following expression:
π
3π
sin x + sin
si ⎛ x + ⎞ + sin(x
sin( x + π) sin
ssi ⎛ x + ⎞ + sin
si ( x +
⎝
⎝
2⎠
2⎠
)
[5 marks]
9C Definition and graph of the
tangent function
We now define another trigonometric function – the tangent
function. It is defined as the ratio between the sine and the
cosine functions.
KEY POINT 9.7
A
N
t
sin x
x
cos x
1
E
B
N
P
t
x
O
sin x
cos x
You may wonder why the function is called the tangent
function. Consider the diagram alongside:
P
O
tan x =
E
A tangent is drawn from the point E, where the unit circle meets
the positive x-axis. The line at angle x is continued until it hits the
tangent. Since OPB and OAE are similar triangles, we get that:
t
i x
=
1 cos x
∴t
x
This can be used to see several important properties of the
tangent function.
As P moves from E to N, the distance along the tangent
increases. When P is close to N, the tangent gets extremely
large. If P is actually at N the line ON is parallel to the tangent
at E so there is no intersection, and tan x is undefined. This
corresponds in the definition to the situation when cos x
is zero, which is not allowed. In fact, the tangent function
π 3π 5π
is undefined for x = , , … These lines are vertical
2 2 2
asymptotes of the graph of y = tan x.
248 Topic 3: Circular functions and trigonometry
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When P moves beyond N the intersection is now below the
x-axis and the value of tan x is negative.
If P ′ is on the opposite side of the circle to P, it still intersects
the tangent at the same point. This means that tan x repeats
every π radians or 180°.
N
P
x
E
O
It is equal to zero when x = 0, , 2π …
t
This information can be used to sketch the graph of y
tan x .
A
KEY POINT 9.8
The graph of the tangent function:
N
y
P
t
x
O
−π
−π2
π
2
π
A
E
x
P
We should remember that the points on the unit circle can also
represent angles measured in degrees.
Asymptotes
are
discussed in Section
4C.
Worked example 9.7
Sketch the graph of y
tan x for −90° < x < 270°.
Find the values for which tan x is
not defined
When is the function positive /
negative?
When is it zero?
© Cambridge University Press 2012
tan x undefined: cos x = 0 when x = −90°, 90°, 270°.
tan x is:
negative in the fourth quadrant,
positive in the first quadrant,
negative in the second quadrant, ...
tan x = 0: x = 0° and 180°
9 Circular measure and trigonometric functions
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249
continued . . .
y
Start by marking the asymptotes
and zeros
−90◦
90◦
180◦
270◦
x
Exercise 9C
1. By estimating the values of sin θ and cos θ, find the
approximate value of tan θ for the points shown on the diagram.
(a)
(c)
(b)
2. Sketch the graph of y
(a) (i) 0° ≤ x ≤ 360°
π
5π
≤x≤
(b) (i)
2
2
tan x for:
(ii) −90° ≤ x ≤ 270°
(ii) − π ≤
≤π
3. Use your calculator to evaluate the following, giving your
answers to 2 decimal places.
(a) (i) tan 1.2
(ii) tan 4.7
(b) (i) tan (–0.65)
(ii) tan (–7.3)
4. Use your calculator to evaluate the following, giving your
answers to 3 significant figures.
(a) (i) tan 32°
(ii) tan168°
(b) (i) tan (–540°)
(ii) tan (–128°)
250 Topic 3: Circular functions and trigonometry
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5. Use the properties of sine and cosine to express the
following in terms of tan x.
π
(a) tan(π − x )
(b) tan ⎛ x + ⎞
⎝
2⎠
(c) tan ( x + )
(d) tan ( x + )
6. Use the properties of sine and cosine to express the
following in terms of tan θ °.
(a) tan (−θ °)
(b) tan (−360 ° − θ °)
(c) tan (−90 ° − θ °)
(d) tan (−180 ° − θ °)
7. Sketch the graph of:
(a) y
π
≤ x ≤ 2π
2
x + tan x for − π ≤ ≤ π
2sin x − tan x for −
(b) y = 3
8. Find the zeros of the following functions:
(a) y =
x ° + si x ° for 0 ≤ x ≤ 360
(b) y =
x ° − tan x ° for −180 ≤ x ≤ 360
9. Find the coordinates of maximum and minimum points
on the following graphs:
(a) y
3sin x − tan x for 0 ≤
(b) y
cos x − 2 sin x for − π ≤
≤ 2π
≤π
10. Find approximate solutions of the following equations,
giving your answers correct to 3 significant figures.
(a) cos x tan x = 3, x ∈] , 2π[
(b) sin x + cos x = 1, x ∈[ , 2π]
9D Exact values of trigonometric
functions
Although values of trigonometric functions are generally
difficult to find without a calculator, there are a few numbers
for which exact values can be found. The method relies on
properties of some special right-angled triangles.
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9 Circular measure and trigonometric functions
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251
Worked example 9.8
π
π
π
Find the exact values of sin , cos and tan .
4
4
4
Mark the point corresponding to
π
on the unit circle
4
A
1
◦
45
s
O
Look at the triangle OAB. It is a right
angle triangle, with the angle
π
at O equal to 45° (because
is
4
one eighth of a full turn)
We can now use the definition of
tan x
s 2 = 1 , so s =
s2
∴ sin
s
B
1
2
π
π
1
= cos =
4
4
2
π
sin
π
4 =1
tan =
4 cos π
4
The other special right-angled triangle is made by cutting an
equilateral triangle in half.
Worked example 9.9
Find the exact values of the three trigonometric functions of
Mark the point corresponding
2π
to
on the unit circle. This is
3
one third of a full turn, so
angle AOB is 60°
252 Topic 3: Circular functions and trigonometry
2π
.
3
A
1
◦
60
B
1
2
O
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continued . . .
Triangle AOB is half of an
equilateral triangle with
side 1. OB is equal to half
the side of the triangle
1
, point A is to the left of the vertical axis, so
2
2π
cos
<0
3
2π
1
∴ cos
=−
3
2
OB =
2
To find AB, use Pythagoras
Use the definition of tan x
3
⎛ 1⎞
2π
3
AB = 1 −
= ∴sin
=
⎝ 2⎠
4
3
2
2
2
3
2π
2
tan
=
=− 3
1
3
−
2
t
exam hin
easier
The table is
if you
r
to remembe
attern
notice the p
s of
in the value
d cos:
both sin an
3 4.
,
0 1, 2 ,
,
2 2
2
2
2
The results for these and other special values are summarised
below. You may be able to remember them all, but you should
understand how they are derived, as shown in the above
examples.
KEY POINT 9.9
Radians
0
π
6
π
4
π
3
π
2
2π
3
3π
4
5π
6
π
Degrees
0
30
45
60
90
120
135
150
180
sin x
0
1
2
2
2
3
2
1
3
2
2
2
1
2
0
cos x
1
3
2
2
2
1
2
0
tan x
0
1
1
3
© Cambridge University Press 2012
3
not
defined
−
1
2
− 3
−
2
2
−1
−
3
2
–1
−
1
0
3
9 Circular measure and trigonometric functions
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253
Exercise 9D
1. By marking the corresponding points on the unit circle, find the
exact values of:
3π
π
5π
3π
(a) cos ⎛ ⎞ (b) cos ⎛ ⎞ (c) sin ⎛ ⎞ (d) tan ⎛ ⎞
⎝ 4⎠
⎝ 2⎠
⎝ 4⎠
⎝ 4⎠
2. Find the exact value of:
4π
π
π
⎛ 7π ⎞
(a) sin ⎛ ⎞
(b) sin ⎝ ⎠ (c) cos ⎛ ⎞ (d) tan ⎛ − ⎞
⎝ 3⎠
⎝ 3⎠
⎝ 6⎠
6
3. Find the exact value of:
(a) cos 45°
(b) sin 135° (c) cos 225° (d) tan 225°
4. Find the exact value of:
(a) sin 210°
(b) cos 210° (c) tan 210° (d) tan 330°
5. Evaluate the following, simplifying as far as possible.
π
π
π
⎛ π⎞
⎛ π⎞
(a) 1 − sin2 ⎛ ⎞ (b) sin ⎛ ⎞ + sin
i ⎛ ⎞ (c) cos ⎝ ⎠ cos ⎝ ⎠
⎝ 4⎠
⎝ 6⎠
⎝ 3⎠
6
6
6. Show that:
(a) sin 60° cos30
cos 30° + cos60
cos ° sin 30° = sin 90°
45°) = 1
(b) (sin 45°) + ( cos 45
π
π
π
i 2 ⎛ ⎞ = cos ⎛ ⎞
(c) cos2 ⎛ ⎞ sin
⎝ 6⎠
⎝ 6⎠
⎝ 3⎠
2
⎛
⎛ π⎞⎞
(d) 1 + tan ⎝ ⎠ = 4 + 2 3
⎝
3 ⎠
2
2
9E Transformations of trigonometric
graphs
See chapter 6 to revise
t ra n s f o r m at i o n s
of graphs.
In this section we shall apply the ideas from chapter 6 to the
trigonometric graphs we have met. It will allow us to model
many real-life situations which show periodic behaviour in
Section 9F and it will also be useful in solving equations in
Section 9G.
Firstly, let us look at the relationship between y sin x and
y sin2 x . The equation y sin2 x is of the form y f ( x ), so
1
we need to apply a horizontal stretch with scale factor to
2
the graph of y sin x .
254 Topic 3: Circular functions and trigonometry
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y
y = sin x
1
y = sin 2x
−π
π
x
2π
−1
We can see that the amplitude of the function is still 1, but the
period is halved to π.
We can compare this to the graph y 2sin x which is of the
form y
f ( x ), so we need to apply a vertical stretch with scale
factor 2 to the graph of y sin x .
y
2
t
exam hin
the
sin 2x is not
n x, and
si
2
same as
sin2x CANNOT be
2
sin x.
simplified to
y = 2 sin x
1
−π
0
y = sin x
π
2π
x
−1
−2
The resulting function has amplitude 2, while the period
is unchanged.
The two types of transformation can be combined to change both the
amplitude and the period of the function. The same transformations
can also be applied to the graph of the cosine function.
KEY POINT 9.10
a sin bx has
2π
amplitude a and period .
b
y
The function y
a cos bx has
2π
amplitude a and period .
b
a
y = a sin bx
The function y
© Cambridge University Press 2012
2π
b
x
−a
9 Circular measure and trigonometric functions
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255
Worked example 9.10
⎛ x⎞
(a) Sketch the graph of y = 4cos ⎜ ⎟ for 0
6π .
⎝ 3⎠
(b) Write down the amplitude and the period of the function.
Start with the graph of y cos x
and think what transformations
to apply to it
(a) Vertical stretch with scale factor 4
Horizontal stretch with scale factor 3
y
y= 4cos (x3)
4
6π
x
−4
t
exam hin
n’ means
‘Write dow
ing is
that no work
required.
(b) amplitude = 4
period =
2π
= 6π
3 −1
As well as vertical and horizontal stretches, we can also
apply translations to graphs. They will leave the period and
the amplitude unchanged, but will change the positions of
maximum and minimum points and the axis intercepts.
Worked example 9.11
(a) Sketch the graph of y
in x + 2 for x ∈[ , 2π].
(b) Find the maximum and the minimum values of the function.
The equation is of the form
y = f(x) + 2. What
transformation does
this give?
(a) Vertical translation by 2 units
256 Topic 3: Circular functions and trigonometry
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continued . . .
y
3
Apply the transformation to the
graph of y = sin x
y = 2 + sinx
2
1
π
−1
We know that the minimum and
maximum values of sin x are 1
and –1. Add 2 to those values
2π
x
y = sin x
(b) Minimum value: −1 + 2 = 1
Maximum value: 1 2 = 3
In the next example we consider a horizontal translation.
Worked example 9.12
(a) Sketch the graph of y
(x +
°) for 0 ≤ x ≤ 360°.
(b) State the minimum and maximum values of the function, and the values of x for which
they occur.
The equation is of the form
y = f (x + 30). What
transformation does this give?
(a) Horizontal translation 30 degrees to the left
y
y = sin x + 30◦
1
Apply the transformation to the
graph of y cosx
180◦
−1
© Cambridge University Press 2012
360◦
x
y = sin x
9 Circular measure and trigonometric functions
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257
continued . . .
The minimum value of cos x is –1,
and it occurs for x 180°, etc. The
graph is shifted 30 units to the left
(b) Minimum value is –1.
It occurs for x = 180 − 30 = 150°
The maximum value of cos x is 1,
and it occurs for x = 0, 360o…..
Pick the value which is in the
required interval after the
translation to the left
Maximum value is 1.
It occurs for x = 360 − 30 = 330°
The result of combining all four transformations can be
summarised as follows.
KEY POINT 9.11
The functions y = asin
a b(x + c) d and y = a cos b(x + c) d
have:
• amplitude a
• minimum value d – a and maximum value d + a
2π
• period
.
b
Notice that the value of d is always half-way between the
min+max
minimum and maximum values. In other words: d =
2
The amplitude is half the difference between the minimum and
max − min
maximum values: a =
2
The value of c in Key point 9.11 determines the horizontal
translation of the graph; therefore it affects the position of
maximum and minimum points. The following example
shows how to work them out.
Worked example 9.13
Find the values of x for which the function y
When does the sine function
attain its maximum?
The given function is of the form
f (3( x 1)) , which means that x
has been replaced by 3(x + 1)
We can now solve for x
i 3( x + 1) has its maximum value.
sin x has a maximum value when x =
3(
x=
258 Topic 3: Circular functions and trigonometry
1) =
π 5π
,
, etc.
2 2
π 5π
,
, etc.
2 2
π
5π
− 1,
− 1, etc.
6
6
© Cambridge University Press 2012
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We can use our knowledge of transformations of graphs to find
an equation of a function given its graph.
Worked example 9.14
The graph shown has equation y = asin(
a
bx ) + d .
y
( π4 , 4)
( 5π4 , 4)
x
0
( 3π4 , −2)
Find the values of a, b and d.
a is the amplitude, which is half the
difference between the minimum
and maximum values
amplitude =
b is related to the period, which is the
distance between the two consecutive
maximum points
2π
The formula is period =
b
period =
d represents the vertical translation
of the graph. It is the value
half-way between the minimum
and the maximum values
d=
© Cambridge University Press 2012
4 − ( −2)
=3
2
∴a = 3
5π π
− =π
4 4
2π
π=
∴b = 2
b
4 + (− )
2
∴d = 1
9 Circular measure and trigonometric functions
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259
Exercise 9E
1. Sketch the following graphs, indicating any axes intercepts.
(a) (i) y sin2 x for − 180° ≤ x ≤ 80°
(ii) y
cos3x for 0° ≤ x ≤ 360°
π
tan ⎛ x − ⎞ for 0 ≤ x ≤ π
⎝
2⎠
π
(ii) y = t ⎛ x + ⎞ for 0 ≤ x ≤ π
⎝
3⎠
(b) (i) y
(c) (i) y
(ii) y
s x − 2 for 0° ≤ x ≤ 7 0°
3
2 i x + 1 for −360° ≤ x ≤ 360°
2. Sketch the following graphs, giving coordinates of maximum
and minimum points.
π⎞
⎛
(a) (i) y cos ⎝ x − ⎠ for 0 ≤ x ≤ 2π
3
π⎞
⎛
(ii) y = i ⎝ x + ⎠ for 0 ≤ x ≤ 2π
2
(b) (i) y
(ii) y
(c) (i) y
(ii) y
2 in ( x + 45°) for − 180° ≤ x ≤ 180°
s ( x − 60°) for − 180° ≤ x ≤ 180°
3
3 in 2 x for − π ≤ x ≤ π
3 2 cos x for 0° ≤ x ≤ 360°
3. Find the amplitude and the period of the following functions:
(a) f ( x ) = 3 i 4 x , where x is in degrees
(b) f ( x ) = tan3x , where x is in radians
(c) f ( x ) = cos3x , where x is in degrees
(d) f ( x ) = 2sin πx , where x is in radians
4. The graph has equation y
values of p and q.
p sin (qx ) for 0 ≤
≤ 2π . Find the
y
5
2π
−5
260 Topic 3: Circular functions and trigonometry
x
[3 marks]
© Cambridge University Press 2012
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5. The graph shown below has equation y
0° ≤ x ≤ 720
7 0°.
a cos( x b) for
Find the values of a and b.
y
2
110
◦
290◦ 470◦
x
650◦
−2
[3 marks]
6. (a) On the same set of axes sketch the graphs of
y 1 + i 2 x and y 2cos x for 0 ≤ ≤ 2π.
(b) Hence state the number of solutions of the equation
1 sin 2 x = 2 cos x for 0 ≤ ≤ 2π.
(c) Write down the number of solutions of the equation
1 sin 2 x = 2 cos x for −2π ≤ ≤ 6π .
[6 marks]
7. (a) Sketch the graph of y
2
( x + °) for x ∈[ °, 360°].
(b) Find the coordinates of the maximum and minimum
points on the graph.
(c) Write down the coordinates the maximum and
minimum points on the graph of
y
( x + °) − 1 for x ∈[ °, 360°].
2
[6 marks]
9F Modelling using trigonometric
functions
In this section we will see how trigonometric functions can be
used to model real-life situations which show periodic behaviour.
Imagine a point moving with constant speed around a circle
of radius 2 cm, starting from the positive x-axis and taking
3 seconds to complete one full rotation. Let h be the height of
the point above the x-axis. How does h vary with time?
2
h
2
−2
θ
2
h
2
3
−2
6
t
−2
© Cambridge University Press 2012
9 Circular measure and trigonometric functions
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261
We know that if the point is moving around the unit circle, the
height above the x-axis is sinθ, where θ is the angle between the
radius and the x-axis. As the circle has radius 2, the height is
2sinθ.
We now need to find how θ depends on time. As the point starts
on the positive x-axis, θ = 0 when t = 0. After one complete
rotation, θ = 2π and t = 3 (t is time measured in seconds).
Because the point is moving with constant speed, we can use
2π
θ
t
t.
ratios to find that
= , so θ =
3
2π 3
Therefore the equation for the height in terms of time is
⎛ 2π t ⎞ . The second diagram above shows the graph
h
⎝ 3 ⎠
of this function.
This is an example of modelling using trigonometric functions.
Sine and cosine functions can be used to model periodic
motion, such as motion around a circle, oscillation of a particle
attached to the end of a spring, water waves, or heights of tides.
In practice, we would collect experimental data to sketch a graph
and then use our knowledge of trigonometric functions to find its
equation. We can then use the equation to do further calculations.
Worked example 9.15
The height of water in the harbour is 16 m at high tide, and 12 hours later at low tide, it is 10 m.
The graph below shows how the height of water changes with time over 24 hours.
h
(a) Find the equation for height (in metres)
16
in terms of time (in hours) in the form
h = m + acos(bt).
(b) Find the first two times after the high tide
10
when the height of water is 12 m.
0
From the previous section: m is
half-way between minimum and
maximum values, a is the amplitude, b
is related to the period
(a) m =
262 Topic 3: Circular functions and trigonometry
12
24
t
16 + 10
= 13
2
© Cambridge University Press 2012
Not for printing, sharing or distribution.
continued . . .
The amplitude is half the distance
between the minimum and
maximum values
The period is
2π
b
Write down the equation for height
Set h = 12
16 − 10
amplitude =
=3
2
∴a = 3
period = 24
2π
24 =
b
So h
∴b =
π
12
π
3 3 co ⎛ t ⎞
⎝ 12 ⎠
⎛π ⎞
(b) 13 + 3 cos ⎝ t ⎠ = 12
12
h
The high tide is when t = 0, so we want
the first two answers with t > 0
12
Solve for t using GDC
7.3
Answer the question
12
16.7
24
t
The height of the water will be 12 m
7.3 hours and 16.7 hours after the high tide.
Exercise 9F
1. The graph shows the height of water below the level of a
walkway as a function of time. The equation of the graph is
of the form y = a cos(bt ) + m. Find the values of a, b and m.
y
6
3
0
12
© Cambridge University Press 2012
24
x
9 Circular measure and trigonometric functions
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263
2. The depth of water in a harbour varies and is given by the
⎛ π t ⎞ , where d is measured in metres
equation d
⎝ 12 ⎠
and t in hours after midnight.
You will see how to
solve equations like
these without a calculator in Section 10B.
(a) Find the depth of the water at low and high tide.
(b) A boat can only enter the harbour when the depth of water
is at least 19 m. Find the times between which the boat can
enter the harbour.
[6 marks]
y
3. A point moves around a vertical circle of radius 5 cm, as shown
in the diagram. It takes 10 seconds to complete one revolution.
5
h
x
(a) The height of the point above the x-axis is given by
h a sin(kt ), where t is time measured in seconds. Find the
values of a and k.
(b) Find the times during the first revolution when the point is
3 cm below the x-axis.
[6 marks]
4. A ball is attached to one end of an elastic string, with the
other end held fixed above ground. When the ball is pulled
down and released, it starts moving up and down, so that the
height of the ball above the ground is given by the equation
h
t, where h is measured in cm and t is time in
seconds.
(a) Find the least and greatest height of the ball above ground.
(b) Find the time required to complete one full oscillation.
(c) Find the first time after the ball is released when it reaches
the greatest height.
[8 marks]
9G Inverse trigonometric functions
To solve equations it will be important that we can ‘undo’
trigonometric functions.
If you were told that the sine of a value is 1 you now know that
2
the original value might be π , but if you knew that the sine of a
6
Inverse
functions
and their graphs
were covered in
Section 5E.
value was 0.8 the original value would not be so easy to find.
We need to find the inverse function of sine. Not every function
has an inverse function; we need the original function to be
one-to-one. Looking at the graph of the sine function, it is clear
that it is not one-to-one. So how can we define inverse sine?
264 Topic 3: Circular functions and trigonometry
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y
− π2
π
2
y = sin x
x
We need to consider the sine function on a restricted domain
where the function is one-to-one. From the graph above we
π
π
can see that a suitable domain is − ≤ x ≤ . (There are other
2
2
options but this is chosen by convention.) It is now possible
to define the inverse function, called the arcsine and written
arcsin x. The graph of the inverse function is the reflection of
the graph of the original function in the line y x. From the
graph alongside we can see the domain and range of the inverse
function.
y
y = arcsin(x)
π
2
y = sin(x)a
1
− 2π
−1
1
π
2
x
−1
−π2
KEY POINT 9.12
The inverse function of f x ) = sin x is f
x ) = arcsin x
π π
or sin −1 x. Its domain is [ , 1] and its range ⎡⎢− , ⎤⎥.
⎣ 2 2⎦
We carry out similar analysis for cosine and tangent functions
to identify the domains on which their inverse functions are
defined. The results are as follows:
KEY POINT 9.13
The inverse function of f x ) = cos x is f
or cos–1x.
x ) = arccos x
t
exam hin
lso
Arcsine is a
n–1 .
known as si
usually
Calculators
a
do not have
lled
button labe
the
arcsin; use
tton
sin−1 bu
instead.
Its domain is [–1, 1] and its range [0, π].
y
π
y = arccos(x)
1
y = cos(x)
−1
1
π
x
−1
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265
KEY POINT 9.14
The inverse function of f x ) = tan x is f
or tan–1x.
⎛ π π⎞
Its domain is R and its range − , .
⎝ 2 2⎠
y
y = tan(x)
y = arctans(x)
π
2
− 2π
x ) = arctan x
0
π
2
x
− π2
You should remember what happens when we compose a
function with its inverse: f f − ( x ) x. Applying this to
inverse trigonometric functions, we get:
(
Inverse
functions
were introduced in
chapter 5.
)
sin(arcsin x ) x
cos(arccos ) x
tan (arctan x ) x
We need to be more careful when composing the other way
round: arcsin(sin x ) x only when x is in the restricted
π π
domain we used to define the arcsin function (so x ∈ ⎡− , ⎤⎥ ).
⎣ 2 2⎦
Worked example 9.16
Solve the equation 3
i (3 ) = arcsin
Evaluate the arcsine and
arccosine terms on the
RHS
Isolate the arcsine
⎛ 1⎞
⎛ 1⎞
+ arccos
.
⎝ 2⎠
⎝ 2⎠
3a c
1
1
(3 ) = arcsin ⎛ ⎞ + arccos ⎛ ⎞
⎝ 2⎠
⎝ 2⎠
π π
+
6 3
π
=
2
=
arcsin(
266 Topic 3: Circular functions and trigonometry
)=
π
6
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continued . . .
Taking sine of both sides undoes
the arcsine
π
sin (arcsin ( 3x )) sin ⎛ ⎞
⎝ 6⎠
1
2
1
x=
6
⇔
3x =
⇔
Exercise 9G
1. Use your calculator to evaluate in radians correct to three
significant figures:
(a) (i) arccos0.6
(ii) arcsin (0.2)
(b) (i) arctan ( − )
(ii) arcsin ( − . )
2. Evaluate in radians without a calculator:
⎛ 3⎞
1
(a) (i) arcsin ⎛ ⎞
(ii) arccos ⎜ ⎟
⎝ 2⎠
⎝ 2 ⎠
(b) (i) arctan(
)
(c) (i) arcsin ( − )
⎛ 1 ⎞
(ii) arccos −
⎝
2⎠
(ii) arctan (1)
3. Evaluate the following, giving your answer in degrees correct to
one decimal place:
(a) (i) arcsin (0.7 )
(b) (i) arccos (−0.. )
(ii) arcsin (0.3)
(ii) arccos (−0.. )
(c) (i) arctan (6.4 )
(ii) arctan (
.1)
4. Find the value of:
(a) (i) sin(arcsin 0.. )
(ii) cos (arccos (
(b) (i) tan (arctan ( − ))
(ii) sin (arcsin ( − ))
5. Find the exact value of:
1 ⎞
⎛
(a) (i) cos arcsin ⎛ ⎞
⎝ 2⎠ ⎠
⎝
⎛
⎛
3⎞⎞
(b) (i) sin arccos ⎜ − ⎟ ⎟
⎝ 2 ⎠⎠
⎝
. ))
⎛
⎛ 2 ⎞⎞
(ii) sin aarccos ⎜ ⎟⎟
⎝ 2 ⎠⎠
⎝
1 ⎞
⎛
(ii) tan arccos ⎛ ⎞
⎝ 2⎠ ⎠
⎝
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267
6. Find the exact value of:
π ⎞
⎛
(a) (i) arcsin sin ⎛ ⎞
⎝ ⎝ 3⎠⎠
2π ⎞
⎛
(b) (i) arcsin ssin ⎛ ⎞
⎝ ⎝ 3 ⎠⎠
5π ⎞
⎛
(ii) arccos cos ⎛ ⎞
⎝ ⎝ 6 ⎠⎠
(ii) arccos (cos ( −
))
7. Solve the following equations:
π
5π
(a) arcsin x =
(b) arccos2 x =
3
6
π
(c) arctan (3 1) = −
6
arcsin x
8. (a) Show by a counterexample that arctan x ≠
.
arccos x
(b) Express arcsin x in terms of arccos x.
(c) Solve the equation 2arctan x
a csi x + arccos x. [8 marks]
arcsin
9. A system of equations is given by:
sin x + cos y = 0.6
cos x sin y = 0.2
(a) For each equation, express y in terms of x.
(b) Hence solve the system for
π
π π
π
− ≤x≤ − ≤y≤ .
2
2
2
2
[8 marks]
Summary
•
The unit circle is a circle with centre O and radius 1.
•
The radian measure is defined in terms of the distance travelled around the unit circle, so
that a full turn = 2π radians.
•
To convert from degrees to radians, divide by 180 and multiply by π.
•
To convert from radians to degrees, divide by π and multiply by 180.
•
Using the unit circle and the real number α, the sine and cosine of this number is defined
in terms of distance to the axes:
sin α is the distance of a point from the horizontal axis
cos α is the distance of the point from the vertical axis
•
Useful properties of the sine and cosine function are summarised in Key point 9.3.
•
The relationship between the sine and cosine function is summarised in Key point 9.4.
•
The tangent function is another trigonometric function. It is defined as the ratio between
sin x
the sine and cosine functions: tan x =
cos x
268 Topic 3: Circular functions and trigonometry
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•
The sine, cosine and tangent trigonometric functions can be defined for all real numbers.
y
y
cosx
sinx
1
x
y
1
y = sin(x)
2π
y = cos(x)
x
2π
y = tan(x)
x
π
2
π
x
3π
2
−1
−1
•
For some real numbers the three functions have exact values, which should be learnt. These
are provided in Key point 9.9.
•
The sine and cosine functions are periodic with period 2π and amplitude 1.
period: sin x
cos x
) = sin(x
ssin(
i ((x ± 4 π) = …
s (
sin(
cos((
) = cos(x
cos(( ± 4 π) = …
amplituted: −1 ≤ sin ≤ 1
d −11 ≤ cos x ≤ 1
π
The cosine graph is a translation of the sine graph units to the left.
2
• The tangent function is periodic with period π.
tan(x)) tan(
t (
•
) = tan(x
ttan(( ± 2π) = …
We can apply transformations to sine and cosine functions.
y = a sin b(x + c) d
y = a cos b(x + c) d have:
2π
amplitude a and period
b
minimum value d a and maximum value d a.
•
d
We introduced inverse trigonometric functions:
Inverse function
arcsin x
Domain
Range
[ , 1]
⎡ π π⎤
⎢⎣− 2 , 2 ⎥⎦
[ , 1]
[ , π]
R
⎛ π π⎞
− ,
⎝ 2 2⎠
(sin −1 x )
arccos x
(cos −1 x )
arctan x
(cos −1 x )
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269
Introductory problem revisited
The original Ferris Wheel was constructed in 1893 in Chicago. It was just over
80 m tall and could complete one full revolution in 9 minutes. During each
revolution, how long did the passengers spend more than 5 m above ground?
A car on the Ferris Wheel moves in a circle. Its height above ground can therefore be
modelled by sine or cosine function.
If the height of the wheel is 80 m then the radius of the circle is 40 m. A car starts on the
ground, climbs to the maximum height of 80 m and returns to the ground. This takes
9 minutes. We can sketch the graph showing the height of the car above the ground as a
function of time.
h
The period of the function is 9, so the equation will involve
⎛ 2π ⎞
⎛ 2π ⎞
either 40 sin ⎝ t ⎠ or 40 cos ⎝ t ⎠ . The amplitude is 40 and the
9
9
graph has been translated up by 40 units. We can choose whether
to use the sine or the cosine function so use the cosine function, as
then there is no horizontal translation involved.
80
0
9
t
Combining all the transformations gives the equation for height in
terms of time.
⎛ 2π t ⎞
⎝ 9 ⎠
h
To find how long the car spends more than 50 m above ground, we need to find the times
when the height is exactly 50 m. This involves solving an equation.
2π
40 − 40 cos ⎛ t ⎞ = 50
⎝ 9 ⎠
This can be solved on a calculator.
t1
2 61
d t2
6.39
Therefore the time spent more than 50 m above ground is
6.39 2.61 ≈ 3.8 minutes.
270 Topic 3: Circular functions and trigonometry
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Mixed examination practice 9
Short questions
1. The height of a wave at a distance x metres from a
buoy is modelled by the function:
f ( x ) = 1.4 sin(3x − 0.1) − 0.6
(a) State the amplitude of the wave.
(b) Find the distance between consecutive peaks of
the wave.
[4 marks]
2. Sketch the graph of y i (2x)
x ) + sin( x ) and hence
find the exact period of the function.
[4 marks]
t
exam hin
ination
Many exam
mbine
questions co
this
ideas from
h further
it
w
r
chapte
y
trigonometr
you will
s
e
techniqu
pters
meet in cha
10–12.
3. A runner is jogging around a level circular track. His distance north of the
centre of the track in metres is given by 60 cos 0.08t where t is measured in
seconds.
(a) How long does is take the runner to complete one lap?
(b) What is the length of the track?
(c) At what speed is the runner jogging?
[7 marks]
π⎞
⎛
4. Let f ( x ) = 3 i 2 x − .
⎝
3⎠
(a) State the period of the function.
(b) Find the coordinates of the zeros of f xx) for x ∈[ , 2π].
(c) Hence sketch the graph of y f ( x ) for x ∈[ , 2π], showing the
coordinates of the maximum and minimum points.
[7 marks]
y
5. The diagram shows the graph of the
function f ( x ) = a s (bx ). Find the
values of a and b.
[4 marks]
(2, 5)
0
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9 Circular measure and trigonometric functions
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271
Long questions
y
1. The graph shows the function f ( x ) = sin( x
A
k ) + c.
(a) (i) Write down the coordinates of A.
(ii) Hence find the values of k and c.
(b) Find all the zeros of the function in the interval [
0
2π
4π
3
, 0].
(c) Consider the equation f ( x ) = k with −0 5 < k < 0.
(i) Write down the number of solutions of this equation in the interval [ , 9π].
(ii) Given that the smallest positive solution is α, write the next two
[11 marks]
solutions in terms of α.
2. (a) (i) Sketch the graph of y tan x for 0 ≤ ≤ 2π.
(ii) On the same graph, sketch the line y π x.
(b) Consider the equation x + t x = π. Denote by x0 the solution of this
π
2
equation in the interval ] , [.
(i) Find, in terms of x0 and π, the remaining solutions of the equation in
the interval [ , 2π].
(ii) How many solutions does the equation x + t
(c) Given that cos A
c and sin A
x = π have for x ∈R?
s,
⎛ π − A⎞ and sin ⎛ π − A⎞ .
⎝2
⎠
⎝2
⎠
π
1
⎛ − A⎞ =
.
(ii) Hence show that tan
⎝2
⎠ tan A
π
4
(iii) Given that tan A ttan
, find the possible values of tan A.
an ⎛
A⎞ =
⎝2
⎠
3
π
(iv) Hence find the values of x ∈] , [ for
2
π
4
tan ⎛
A⎞ =
.
[16 marks]
which tan A tan
⎝2
⎠
3
(i) Write down the values of cos
3. (a) Write down the minimum value of cos x and the smallest positive value of x
(in radians) for which the minimum occurs.
(b) (i) Describe two transformations which transform the graph of y cos x to
the graph of y = 2
⎛ x + π⎞.
⎝
6⎠
⎛
⎝
(ii) Hence state the minimum value of 2 cos x +
x ∈[
] for which the minimum occurs.
(c) The function f is defined for x ∈[
π⎞
and find the value of
6⎠
] by f ( x ) =
5
π
2 cos ⎛ x + ⎞ + 3
⎝
6⎠
(i) State, with a reason, whether f has any vertical asymptotes.
(ii) Find the range of f .
272 Topic 3: Circular functions and trigonometry
.
[13 marks]
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x