SECTION 8.4: TRIG IDENTITIES

& EQUATIONSPre-Calculus

8.4 TRIG IDENTITIES & EQUATIONS

Objectives:1. Identify the relationship of trig functions

and positive and negative angles2. Identify the Pythagorean trig relationships3. Identify the cofunction trig relationships4. Apply various trig relationships to simplify

expressions.

Vocabulary:sine, cosine, tangent, cosecant, secant, cotangent, cofunction

REVIEW OF RECIPROCAL TRIG RELATIONSHIPS

EXAMPLE 1: SIMPLIFYING EXPRESSIONS Simplify the following Expressions

PART 1:PYTHAGOREAN TRIG RELATIONSHIPS

Let’s take a look at the unit circle. Using the Pythagorean Theorem, how

can you relate all three sides of the triangle?sin2θ + cos2θ = 1

This is one of the PythagoreanTrig Relationships

EXAMPLES: SIMPLIFYING EXPRESSIONS

PART 1:PYTHAGOREAN TRIG RELATIONSHIPS

Starting with sin2θ + cos2θ = 1, how can you manipulate it to get other following Pythagorean Trig Relationships?

1 + tan2θ = sec2θ Divide both sides by cos2θ

1 + cot2θ = csc2θ Divide both sides by sin2θ

These are the final 2 of the 3 Pythagorean Trig Relationships

EXAMPLES: SIMPLIFYING EXPRESSIONS

PART 2:COFUNCTION TRIG RELATIONSHIPS Sine & Cosine, Tangent & Cotangent,

Secant & Cosecant are all Cofunctions. Trig Cofunctions have the following

relationship

WHY?

EXAMPLES: SIMPLIFYING EXPRESSIONS Simplify the following

tan (90° – A) =

Cos (π/2 – x) =

PART 3:TRIG RELATIONSHIPS WITH NEGATIVE & POSITIVE ANGLES

Let’s take a look at a positive and negative angle on the unit circle

PART 3:TRIG RELATIONSHIPS WITH NEGATIVE & POSITIVE ANGLES

Let’s take a look at sin θ. What does this equal according to our picture?

What about sin –θ. What does this equal according to our picture?

What can we say about the relationship between sin θ & sin –θ?

PART 3:TRIG RELATIONSHIPS WITH NEGATIVE AND POSITIVE ANGLES

We just proved that sin (-θ) = - sin θ What do you think the relationship

between cos (- θ) and cos θ is?cos (- θ) = cos θ

What about the relationship between tan (- θ) and tan θ? tan (- θ) = - tan θ

Let’s look at csc (- θ) and csc θ. What is the relationship? csc (- θ) = - csc θ

What about the relationship betweensec (- θ) and sec θ? sec (- θ) = sec θ

What about the relationship between cot (- θ) and cot θ? cot (- θ) = - cot θ

Part 3:Trig Relationships With Negative and Positive Angles

EXAMPLES: PRACTICE SIMPLIFYING Write the equivalent trig function with a

positive angle Sin (-π/2)

Cos (-π/3)

Cot (-3π/4)

SUGGESTIONS Start with the more complicated side Try substituting basic identities (changing all

functions to be in terms of sine and cosine may make things easier)

Try algebra: factor, multiply, add, simplify, split up fractions

If you’re really stuck make sure to:

Change everything on both sides to sine and cosine.

Work with only one side at a time!

DON’T GET DISCOURAGED! Every identity is different Keep trying different approaches The more you practice, the easier it will be

to figure out efficient techniques If a solution eludes you at first, sleep on it!

Try again the next day. Don’t give up! You will succeed!

TIPS TO HELP SIMPLIFY EXPRESSIONS There are 4 different categories of trig

relationships which each have different key components to look forReciprocal Relationships

Most commonly used in some type of format similar to cot y · sin ymanipulating a fraction with trig functions

Usually the functions aren’t squared when they are in this format

Negative/Positive Angle Relationships Similar to the example problems previously in

this powerpoint tan (-45°)

TIPS TO HELP SIMPLIFY EXPRESSIONS There are 4 different categories of trig

relationships which each have different key components to look for Cofunction Relationships

Similar to the example problems previously in this powerpoint

cos (90° – A) Pythagorean Relationships (MOST

COMMON/CHALLENGING!) Includes exponents to the second degree Includes expanding two binomials Addition and subtraction of fractions May need to factor out a trig function before

simplifying Or some type of variation of the previous

TIPS TO HELP SIMPLIFY EXPRESSIONS Though most of the problems are

separated into their respective categories, you may find yourself having to combine multiple relationships to fully simplify an expression.Maybe you’ll start with Pythagorean

relationships, then to fully simplify you may use Reciprocal relationships.

In most cases, fully simplifying an expression will leave the expression with only one term

HOMEWORK Textbook pg 321: #1, 5, 13, 21, 31