Section 8.4: Trig Identities & Equations

Pre-Calculus

Learning TargetsReview Reciprocal Trig RelationshipsExplain the relationship of trig functions with

positive and negative anglesExplain the Pythagorean trig relationshipsExplain the Cofunction trig relationshipsApply various trig relationships to simplify

expressions

Review of Reciprocal Trig Relationships

Example 1: Simplifying ExpressionsSimplify the following Expressions

Part 1:Trig Relationships with Negative & Positive AnglesLet’s first take a look at a positive and

negative angle on the unit circle

Part 1:Trig Relationships with Negative & Positive AnglesLet’s take a look at What does this

equal according to our picture?What about What does this equal

according to our picture?What can we say about the relationship

between

Part 1:Trig Relationships With Negative and Positive AnglesWe 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 between sec (- θ) and sec θ? sec (- θ) = sec θ

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

Part 1:Trig Relationships With Negative and Positive Angles

Examples: Practice SimplifyingWrite the equivalent trig function with a

positive angleSin (-π/2)

Cos (-π/3)

Cot (-3π/4)

Part 2:Pythagorean Trig RelationshipsLet’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

Part 2:Pythagorean Trig RelationshipsStarting with sin2θ + cos2θ = 1, how can you

manipulate it to get other following Pythagorean Trig Relationships?

1 + tan2θ = sec2θDivide both sides by cos2θ

1 + cot2θ = csc2θDivide both sides by sin2θ

These are the final 2 of the 3 Pythagorean Trig Relationships

Examples: Simplifying Expressions

Part 3:Cofunction Trig RelationshipsSine & Cosine, Tangent & Cotangent, Secant

& Cosecant are all Cofunctions.Trig Cofunctions have the following

relationship

The relationships still hold if the angle is in radians (π/2)

Examples: Simplifying ExpressionsSimplify the following

tan (90° – A) =

Cos (π/2 – x) =

Tips to help simplify expressionsThere 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 y manipulating 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 expressionsThere are 4 different categories of trig relationships

which each have different key components to look forCofunction 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 expressionsThough 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

HomeworkTextbook pg 321: #1, 12, 13, 19