Trigonometric Identities and Cofunction Identities Guided Exploration
Activity 1: Using a Right Triangle to determine equivalent trigonometric expressions
1. Draw a right angled triangle ABC where C is the right angle.
Label the measure of angle A as x. Label the sides a, b and c.
2. Write an expression for angle B in terms of x and
Draw triangle here:

.
2
3. Fill in the blanks _______ using your expression for B and then
complete the equations using the lengths a, b and c.
sin ______ =
cos _______ =
tan _______ =
4. Complete the equations using the lengths a, b and c.
sin x =
cos x =
tan x =
5. Which expressions are equal? These are equivalent trigonometric expressions and are called trigonometric
identities. Write them into the Trig Identity table on page 2 of this note. The first one has been filled in for you.
An identity is an equation that is true for all values of the variable for which the expressions on both sides of the
equation are defined. An identity involving trigonometric expressions is called a trigonometric identity.
6. Why are these expressions equal?
Cofunction Identities are made up of complementary angles. In the next activity we will look at the identities created
with supplementary angles. (Complementary angles add to a right angle, supplementary angles add to a straight line).
Activity 2: Using the Unit Circle and transformations to determine equivalent trigonometric expressions
1. Draw any angle x in the first quadrant using a ruler and compass.
Label the end of the terminal arm P and the origin O.
Drop a vertical line to create a right triangle POQ.
Label P with its coordinates in terms of sine and cosine.
Draw angle on this graph:
2. Rotate point P 90o (add a right angle) to create point P’
(read P prime, means the translation of point P).
Your terminal arm should now make an angle of x 

2
with the initial arm.
3. What is the measure of the reference angle? Create a new right triangle P’OQ’.
4. Why is POQ  P'OQ ' ?
5. What are the coordinates of P’ in terms of sin x and cos x? Use the coordinates of P’ to help you identify all of
the equivalent trigonometric expressions for x and x 
table.

2
. Record these new trigonometric identities in the
Trig Identities identified so far:


sin x  cos  x 
2

There are other identities using  ,
3
and 2 . If there is time at the end of class we will find these 18
2
identities.
Activity 3: Why are identities useful?
Expressions that arise from real life situations are generally messy and can sometimes be simplified using trig
identities. Note that you can use graphing technology to demonstrate the two trigonometric expressions are
equivalent by graphing them both and verifying that they overlap.
What are you going to ask me to do with them?
You will be asked to substitute, simplify and solve using these and other trig identities. For example:
Given sin

6

1
, use an equivalent
2
trigonometric expression to show that
cos
2
1
 .
3
2
Given sin

5
 0.5878 , find cos
Given cos

7
 sin y , first express
difference between

as a
7

and an angle, and then
2
apply a cofunction identity to determine the
measure of angle y.
7
.
10
Given csc
2
3
 1.2790 , find sec
.
7
14