1. Lesson 3

1.8. Explore 4

Mathematics 30-1 Module 5

Module 5: Trigonometry Applications and Identities

 

In Try This 4 you may have found that sin θ and cos θ are related by the identity sin2 θ + cos2 θ = 1 by drawing a diagram similar to the diagram shown. You also saw that 1 + tan2 θ = sec2 θ and 1 + cot2 θ = csc2 θ. As such, they are identities. These are called Pythagorean identities because each can be derived from the Pythagorean theorem, a2 + b2 = c2.

 

This diagram shows a unit circle with a terminal arm at angle theta. A right triangle is drawn using the point the terminal arm intersects the circle, the point on the x-axis directly below this point, and the origin. The hypotenuse is labelled with 1 and c, the side along the x-axis is labelled with cosine theta and a, and the vertical side is labelled sine theta and b.

 

tip
It is often useful to rearrange an identity to suit your needs. sin2 θ + cos2 θ = 1 can be rewritten as sin2 θ = 1 − cos2 θ or as cos2 θ = 1− sin2 θ. Pay attention to any restrictions on the identity if you rearrange it.

Watch “Pythagorean Identity Derivation” to see another derivation of these identities.

 

 
This play button opens �Pythagorean Identity Derivation.�
Source: Khan Academy (CC BY-NC-SA 3.0)

 

 



textbook

Read “Example 3” on page 295 of the textbook. Pay attention to how the identity is verified numerically.

 

 

 

Self-Check 3
  1. Complete the activity in “Simplifying Trigonometric Expressions."

     
    This play button opens �Simplifying Trigonometric Expressions.�

 

textbook

  1. Complete questions 4.c., 7, 11, 13, and 14 on pages 296 to 298 of the textbook. Answer