8.4 Relationships Among the Functions

Objectives:

1. Simplify trig expressions

2. Prove trig identities

Relationships with Negatives

If P is a point onthe unit circle, then x = cos θand y = sin θ.

Pythagorean Identities

Since P(cos θ, sin θ) is on the unit circle:

We’ll verify these later!

To Simplify:

Use trig identities to try to express as one trig function or as a constant.

It often helps to rewrite everything in terms of sine and cosine.

Example:

Simplify the expression.

Example:

Simplify the expression.

Example:

Simplify:

You Try!

Simplify each expression:

Investigating Cofunctions

Use your calculator to evaluate:

sin 50° cos 40°

sin 25° cos 65°

sin 79° cos 11°

sin 7° cos 83°What do you notice?

Why Does This Happen?

Cofunction Relationships

Proving Identities

Simplify one (or both) side(s) of the equation to show they are equal.

Need to work with each side separately. Don’t move anything across the = sign.

Example:

Prove:

Example:

Prove:

You Try!

Prove: