CHAPTER 3
LESSON 3
Teacher’s Guide
The Sine and Cosine Functions of Angles in Standard Position
AW 3.4
MP 4.2
Objectives:
• To introduce the unit circle
• To introduce sine and cosine as circular functions
• To introduce the sine and cosine of angles in standard position using a unit circle
and r
Prerequisites
1. Consider the right angle triangle below.
(i) State the opposite side of the angle θ .
(ii) State the adjacent side of the angle θ .
(iii) Identify the hypotenuse.
(iv) Apply the Pythagorean Theorem.
(i) opposite side:
(iii) hypotenuse:
a
c
(ii) adjacent side:
b
(iv) Pythagorean Theorem: a2 + b2 = c2
2. Using the above triangle, specify the sine and cosine of the angle θ .
side opposite
sinθ = hypotenuse = ac
adjacent b
cosθ = side
hypotenuse = c
These are the classical Greek definitions of sine and cosine.
1. Introducing the Unit Circle
The unit circle is simply any circle whose radius equals ____1_____.
For our purposes, we will place the centre
of a unit circle at the origin of the Cartesian
plane.
2. Defining Sine and Cosine as Circular Functions in a Unit Circle
The classic Greek definitions for sine and cosine hold for acute angles only: i.e.,
0≤θ≤ π
2.
We will now extend the definitions of sine and cosine to include all angles.
Let θ be any angle in standard position.
Let P(x,y) be the point where the terminal
arm of θ intersects the unit circle.
We define the sine and cosine of the angle
θ as follows.
sinθ = y
cosθ = x
This is consistent with the classical definitions involving side opposite, side adjacent and
hypotenuse. (SOHCAHTOA)
sinθ =
side opposite y
=
1
hypotenuse
cosθ =
side adjacent x
=
hypotenuse
1
Thus, we can re-label the coordinates of P as follows:
Note that with this definition, we are not restricted to the sine and cosine of acute angles.
Regardless of the size of θ , its terminal side will always intersect the unit circle at some
point P( x , y ) in one of the four quadrants.
Example 1:
The terminal arm of an angle θ intersects the unit circle at the point (– 0.6, 0.8). Sketch
θ (and its reference angle θr ) on the diagram below. (Use the dot as your guide.) State
the sine and cosine of θ as well as the sine and cosine of θr .
By the circular definitions of sine and
cosine:
sinθ = y = 0.8
cosθ = x = −0.6
By the classical definitions of sine and
cosine (SOHCAHTOA):
opp 0.8
sinθ r =
=
= 0.8
hyp
1
cosθ r =
adj 0.6
=
= 0.6
hyp
1
Note:
1) The sine of any quadrant II angle is always _____positive_____, since the y–
coordinate of P is always positive in that quadrant.
2) The cosine of any quadrant II angle is always _____negative____, since the x–
coordinate of P is always negative in that quadrant.
3) In quadrant II, how are the sine and cosine of an angle θ related to the sine and cosine
of its reference angle?
For θ in quadrant II, we have
sinθ = sinθr
but cosθ = −cosθr
Example 2:
On the grid below, sketch the angle θ = 230° in standard position. (Use the point
provided as a guide.) As well, draw θr the reference angle of 230° .
a) Determine the coordinates of the point P( x , y ) on the terminal arm of θ .
b) Explain the relationship between the sine and cosine of θ and the sine and
cosine of θr .
a) Since the x and y – coordinates of P are
sin230° and cos 230° respectively,
they can be determined from the calculator.
y = sin230 °=⋅ − 0.7660
x = cos 230°=⋅ − 0.6428
b) Note from the diagram below how sin 50° and cos50° relate to sin 230° and cos230°.
opp ⋅ 0.7660 ⋅
sinθ r = sin50° =
=
= 0.7660
hyp
1
cosθ r = cos50 °=
adj ⋅ 0.64278 ⋅
=
= 0.64278
hyp
1
For θ in quadrant III, we have
sinθ = -sinθr
and cosθ = -cosθr
Notes:
1) The sine of any quadrant III angle is always _____negative_____, since the y–
coordinate of P is always positive in that quadrant.
2) The cosine of any quadrant III angle is always _____negative____, since the x–
coordinate of P is always negative in that quadrant.
3. Sine and Cosine of Quadrant Angles
By definition, quadrant angles are standard position angles whose terminal arms
coincide with either the x–axis or the y–axis_____________________________.
In other words, the quadrant angles are 0, π2 , π , 32π , 2 π , 52π , 3π etc.
°
These angles are all integral multiples of π
2 or 90 .
Exercise:
Use the unit circle definitions to compute the sine and cosine of each of the following
quadrant angles.
cos0 = x = 1
sin0 = y = 0
cos3π 2 = x = 0
sin 3π 2 = y = −1
π
cos = x = 0
2
π
sin = y = 1
2
cos2π = x = 1
sin2π = y = 0
cosπ = x = −1
sinπ = y = 0
cos5π 2 = x = 0
sin5π 2 = y = 1
4. The General Circular Definition of Sine and Cosine
Consider an angle θ in standard position, with P( x , y ) any point on the terminal arm of
θ , at a distance r from the origin.
We define sinθ and cosθ as follows.
y
sinθ = r
cosθ = x
r
By Pythagoras, we also have
2
2
2
x +y =r
Note that our previous definition (using the unit circle) is a particular case of the general
definition above (using a circle of radius r). Let P( x1 , y1 ) be the point where the terminal
arm of θ intersects the unit circle.
Figure 1
Figure 2
Note that the triangle formed in the unit circle (Figure 2) is similar to the triangle formed
in the circle of radius r (Figure 1).
Thus, we have the following result.
x x1
y y1
=
=
cos
θ
and
r
r = 1 = sinθ
1
Example 3:
The terminal arm of an angle θ contains the point (–2, –1).
• Sketch θ (and its reference angle θr ) on the grid below.
• Compute the sine and cosine of θ as well as the sine and cosine of θr .
• Find θ to the nearest tenth of a degree.
Note that by Pythagoras:
2
2
2
r = (−2) + (−1) and
r= 5
By the circular definitions of sine and
cosine:
y -1 ×
sinθ = =
= - .4472
r
5
cosθ =
x -2 ×
=
= - 8944
r
5
By the classical definitions of sine and
cosine (SOHCAHTOA) :
opp
1 ×
sinθr =
=
= .4472
hyp
5
cosθr =
adj
2 ×
=
= .8944
hyp
5
Recall that you can use a calculator to determine an angle when you know its sine or
cosine. We will first compute the reference angle of θ .
This means that θr = 26.5° . From the diagram above, we can now find θ itself.
θ = 180° + θ r = 180° + 26.6° = 206.6°
Can you explain the calculator output when you compute sin−1 (−.4772) and
cos−1 (−.8944) directly?
Note to teachers: This last question, while beyond the range of this course, could serve
as an excellent open-ended challenge for your students.
Example 4:
If θ is a fourth-quadrant angle for which cosθ = 12
13 , draw θ and its reference angle on
the grid below, and find sinθ . Last, find θ itself to the nearest thousandth of a radian.
Let P(x , y ) be a point on the terminal side
of θ .
Since cosθ = x
r , let x = 12 and r = 13.
By Pythagoras, we can find y.
x 2 + y2 = r 2
2
2
2
2
2
y = r − x = 13 − 12 = 25
So y = −5. (y < 0 because P is in Quad IV)
Therefore, sinθ =
y −5
=
r 13
By the classical definitions of sine and cosine (SOHCAHTOA) :
opp 5
adj 12
sinθ r =
=
and
cosθ r =
=
hyp 13
hyp 13
To find θ , we first compute its reference angle.
12
–5
θ r = cos−1
= sin−1 13 = .39 47911197 (your choice)
13
From the diagram, we can evaluate θ itself.
θ = 2π – θ r = 2 π − .3947911197 = 5.888
Note:
1) The sine of any quadrant IV angle is always _____negative_____, since the y–
coordinate of P is always negative in that quadrant.
2) The cosine of any quadrant IV angle is always _____positive____, since the x–
coordinate of P is always positive in that quadrant.
3) In quadrant IV, how are the sine and cosine of an angle θ related to the sine and
cosine of its reference angle?
sinθ = -sinθr
and cosθ = cosθr
Summary
The sine and cosine of an angle θ will vary in sign, depending on the quadrant of the
angle.
Fill in the table below, indicating the sign (positive or negative) of both the sine and
cosine according to the quadrant in which θ terminates.
Quadrant I
positive
Quadrant containing θ
Quadrant II
Quadrant III
positive
negative
Quadrant IV
negative
positive
negative
positive
Trig Function
y
sinθ = r
cosθ = xr
negative
In general, the sine and cosine of an angle θ will equal the sine and cosine of its
reference angle respectively, except for a possible change in sign from positive to
negative. The particular change in sign involved (if any) is determined by what quadrant
θ lies in.
Example 5:
Use your calculator to find the sine or cosine of each angle to 4 decimal places. State
which quadrant the angle is in and state why the trig ratio is positive or negative.
a) sin 100º = 0.9848
Quadrant: II
Reasoning: The sine is positive because y
is positive in Quadrant II.
b) cos 250º = –0.3420
Quadrant: III
Reasoning: The cosine is negative
because x is negative in Quadrant III.
c) sin 9π
4 =_0.7071__
Quadrant: I
Reasoning: The sine is positive because y
is positive in Quadrant I.
d) cos 11π
6 = _0.8660_
Quadrant: IV
Reasoning: The cosine is positive
because x is positive in Quadrant IV.