Chapter 3 - Trigonometric Functions
If I turn my car around to face the other direction, we say I have turned it 180 degrees. Degrees
are one of the units we use to measure an angle. We base the degree system off the fact that a
circle has 360°. So, if we’re turning if our angle is one-quarter of a full circle, we say it has
1
measure (360 o ) = 90 o .
4
However, this is not the way most sciences measure angles. Just as we have Fahrenheit and
Celsius measures of temperature, there is more than one way to measure an angle. The preferred
method in the sciences is to work in radians. Almost all of your work in math from this point
forward will work in radians.
To find the measure of an angle using radians, we use the fact that a circle has 2π radians. So, if
we are talking about an angle that measures a quarter of a circle, it’s radian measure would be
1
!
(2! ) = . To convert between degrees and radians, we can use the following:
4
2
& 180 #
1 radian = $
! degrees
% ' "
& ' #
1 degree = $
! radians
% 180 "
Problems
1) Convert the following degrees to radians.
Degr
ees
Radi
ans
30
45
Degr
ees
Radi
ans
60
210
75
225
90
240
105
270
120
315
135
330
150
360
180
Trigonometric functions for acute angles
To begin our conversation about trigonometric functions, remember the Pythagorean theorem
guarantees for any right triangle with sides of length a, b and c, the equation a2+b2=c2 will be
true. Also, remember two famous triangles: the 45-45-90 triangle (the triangle with angle
measures 45°, 45° and 90°) and the 30-60-90 triangle (the triangle with angle measures 30°, 60°
and 90°). Any triangle with the angle measures of 45-45-90 will have sides with lengths
proportional to 1, 1 and 2 . Any triangle with angles measuring 30°, 60° and 90° will have
sides with lengths proportional to 1, 2 and 3 . These triangles with their radian measures are
shown below.
!
3
!
4
2
1
2
1
!
6
!
4
1
3
Since the ratio of the sides of any two similar triangles are the same, we have some functions to
describe these ratios. The sine function represents the ratio of the side opposite θ to the
hypoteneuse, the cosine function is the ration of the side adjacent to θ to the hypotenuse and the
tangent function is the ratio of the side opposite θ to the side adjacent to θ. These formulas can
be remembered as “SOHCAHTOA”.
opp
sin ! =
hyp
adj
cos ! =
hyp
opp
tan ! =
adj
Example 1: Find sin
!
.
6
opp
!
. Since sin measures
we look
hyp
6
!
! 1
at the side opposite
and get 1 and the hypotenuse is 2. Thus, sin = .
6
6 2
Solution: We locate the triangle with an angle measuring
An easy way to remember the values for sine and cosine for these basic acute angles is to draw a
table. List the basic angles in increasing order as follows:
θ
sin θ
cos θ
0
π/6
π/4
π/3
π /2
Then, fill in all the blanks with a fraction with a 2 in the denominator.
θ
sin θ
cos θ
0
/2
/2
π/6
/2
/2
π/4
/2
/2
π/3
/2
/2
π /2
/2
/2
Working your way across the row for sin, fill in with the numerators with 0,1, 2 ,
θ
0
π/6
π/4
π/3
π /2
sin θ
0/2
1/2
2 /2
3 /2
2 /2
cos θ
/2
/2
/2
/2
/2
3 , 2.
Last, fill in the cosine row from right to left with the same numbers: 0,1, 2 , 3 , 2.
θ
0
π/6
π/4
π/3
π /2
sin θ
0/2
1/2
2 /2
2 /2
3 /2
cos θ
2/2
1 /2
0/2
2 /2
3 /2
Problems
2) Fill in the following chart using your knowledge of the triangles. Do not use the above
chart.
!
sin !
cos!
tan !
0
!
6
!
4
!
3
!
2
Trigonometric Functions for Other Angles
The above method will help us evaluate the trigonometric functions for acute angles. To
evaluate obtuse or negative angles, we can apply this information to the circle with radius r. If
we let (x,y) be any point on the circle then the trig functions can be defined as:
(x,y)
y
r
x
cos ! =
r
y
tan ! =
x
sin ! =
r
y
!
x
Note that this gives the same result as the triangles above, it’s just a different way to look at it.
Example 2: Find cos(0).
Solution: We must find the x-coordinate and y-coordinate that correspond to the angle of 0. If
we assume we’re working on a circle of radius 1, then the angle with measure 0 will intersect the
circle at the point (1,0). Thus x=1 and y=1 so cos(0)=1.
Obtuse angles
To evaluate the trigonometric functions for obtuse angles, we use the idea of reference angles.
The reference angle of θ is the measure of the angle between the terminal side of θ and the x7!
!
axis. So, for example, the reference angle of
is
because the measure from the x-axis to
6
6
7!
!
is .
6
6
7!
6
!
6
Evaluating a trig function at any angle can be accomplished by evaluating that function at it’s
reference angle and then adjusting the sign if necessary. Since sine depends on y, it is positive
where y is positive: in the first and second quadrants. Since cosine depends on x, it is positive
where x is positive: in the first and fourth quadrant. Since tangent depends on both x and y, it is
positive when both x and y have the same sign: in the first and fourth quadrants.
This can be remembered as “All Students Take Calculus”.
S
A
Sine
All
T
C
Tangent
Cosine
& 7'
Example 3: Find sin $
% 6
#
!.
"
7!
!
& 7' #
Solution: To evaluate sin $
which is . So,
! , we first find the reference angle of
6
6
% 6 "
& 7' #
&' # 1
instead of finding sin $
! we find sin $ ! = . We then check to be sure the sign is correct.
% 6 "
%6" 2
7!
Since
is in the third quadrant, we know it’s sine value should be negative, so
6
1
' 7( $
sin %
"=! .
2
& 6 #
& 3' #
Example 4: Find tan$ ! .
% 4 "
3!
!
3!
&' #
is . tan$ ! = 1 and
is in the second quadrant
4
4
4
%4"
' 3( $
where tangent is negative. Thus, tan% " = !1 .
& 4 #
Solution: The reference angle of
Problems
3) Fill in the following chart of values using the unit circle, and your knowledge of reference
angles.
!
!
3
2!
3
!
4!
3
5!
3
5!
6
7!
6
11!
6
3!
2
3!
4
5!
4
7!
4
2!
sin !
cos!
tan !
Other Trigonometric Functions
Secant, cosecant and cotangent are related sine, cosine and tangent as follows:
1
csc ! =
sin !
1
sec ! =
cos !
1
cot ! =
tan !
One way to remember which function goes with which is that there is exactly one “co” in each
pair.
& 5' #
Example 5: Evaluate sec$ ! .
% 4 "
& 5' #
& 5' #
Solution: To find sec$ ! , we first find cos$ ! .
% 4 "
% 4 "
1
' 5( $
cos% " = !
2
& 4 #
1
1
& 5' #
sec$ ! =
=
=! 2
% 4 " cos& 5' # ! 1
$ !
2
% 4 "
Problems
4) Fill in the following chart using your knowledge of the triangles. Do not use the above
chart.
!
!
6
!
4
!
3
csc!
sec!
cot !
5) Fill in the following chart of values using the unit circle, and your knowledge of reference
angles.
!
csc!
sec!
cot !
!
3
2!
3
4!
3
5!
3
5!
6
7!
6
11!
6
3!
4
5!
4
7!
4
Graphs of the trigonometric functions
The graphs of the basic trig functions are given below.
Inverse trigonometric functions
Sometimes we want to work a trigonometric function backwards. So, given a ratio of sides, we
want to find the angle. To do this, we use the inverse trig functions. Remember that for a
function to have an inverse it must pass the horizontal line test. So, in order to make our trig
functions invertible we restrict their domains.
sin-1x=y if and only if siny=x and –π/2<y< π /2
cos-1x=y if and only if cosy=x and 0<y< π
tan-1x=y if and only if tany=x and –π/2<y< π /2
Example 6: Find tan-1(1).
Solution: This asks what angle gives a tangent value of 1?
π/4
Problems
1) Find the following:
1
a) sin !1
2
b) sin !1 0
c) sin !1
2
2
d) sin !1
3
2
& 1#
e) sin '1 $ ' !
% 2"
&
3#
!
f) sin '1 $$ '
!
2
%
"
g) cos !1
h) cos !1
3
2
2
2
&
i) cos '1 $$ '
%
&
j) cos '1 $$ '
%
m) tan !1
3
3
n) tan !1 1
2#
!
2 !"
o) tan !1 0
3#
!
2 !"
p) tan !1 (!1)
k) cos !1 0
q) tan !1 (! 3)
l) cos !1 1
&
3#
!
r) tan '1 $$ '
!
3
%
"
g) Trigonometric Identities
In working with trigonometric expressions, there are times when we want to move from one
1
expression to one that is easier to work with. For example, we might have
in an equation
sec x
and we can agree that cosx is a much simpler way to express this fraction.
Here are some of the most common identities you’ll be using:
Pythagorean Identities
sin2x+cos2x=1
1+tan2x=sec2x
1+cot2x=csc2x
The last two from above can be remembered using the sentence “I tan in a second.”
Even and Odd Identities
sin(-x) = -sinx
cos(-x) = cosx
Since we can take the negative sign and write it in front of the sine function, we say sin is odd.
Since the putting a negative angle into the cosine function is no different from the positive angle
we say cosine is even.
Angle addition formulas
Please remember that sin(a+b)≠sin(a)+sin(b). Instead you must work it out using the following
formulas:
sin(a+b)=sin(a)cos(b) + cos(a)sin(b)
cos(a+b)=cos(a)cos(b) – sin(a)sin(b)
! !
+ ).
3 4
3& 2# 1& 2#
6+ 2
! !
!
!
!
!
$
!+ $
!=
Solution: sin( + ) = sin( ) cos( ) + cos( ) sin( ) =
$
!
$
!
3 4
3
4
3
4
2 % 2 " 2% 2 "
4
Example 7: Find sin(
Example 8: Simplify
Solution:
1 ! sin 2 x
cos(! x)
1 ! sin 2 x cos 2 x
cos 2 x
=
= cos x
=
cos(! x) cos(! x) cos( x)
Problems
2) Verify each identity.
a) cos x csc x = cot x
b)
cos x sec x
= tan x
cot x
c) tan r +
d)
cos r
= sec r
1 + sin r
cos t
1 ! sin t
+
= 2 sec t
1 ! sin t
cos t
3) Find the exact value of each expression.
&' ' #
a) sin $ + !
%4 6"
& 3' ' #
+ !
b) cos$
% 4 6"
&' ' #
c) sin $ + !
%6 3"
& 4' ' #
+ !
d) cos $
4"
% 3
e) cos(75 o )
f) sin(105 o )