Section 8 4 Trig Identities Equations PreCalculus Learning

Section 8. 4: Trig Identities & Equations Pre-Calculus

Learning Targets �Review Reciprocal Trig Relationships �Explain the relationship of trig functions with positive and negative angles �Explain the Pythagorean trig relationships �Explain the Cofunction trig relationships �Apply various trig relationships to simplify expressions

Review of Reciprocal Trig Relationships � � �

Example 1: Simplifying Expressions �Simplify the following Expressions � �

Part 1: Trig Relationships with Negative & Positive Angles �Let’s first take a look at a positive and negative angle on the unit circle

Part 1: Trig Relationships with Negative & Positive Angles �Let’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 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

Part 1: Trig Relationships With Negative and Positive Angles �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

Examples: Practice Simplifying �Write the equivalent trig function with a positive angle �Sin (-π/2) �Cos (-π/3) �Cot (-3π/4)

Part 2: 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? �sin 2θ + cos 2θ = 1 �This is one of the Pythagorean Trig Relationships

Part 2: Pythagorean Trig Relationships �Starting with sin 2θ + cos 2θ = 1, how can you manipulate it to get other following Pythagorean Trig Relationships? � 1 + tan 2θ = sec 2θ � Divide both sides by cos 2θ � 1 + cot 2θ = csc 2θ � Divide both sides by sin 2θ �These are the final 2 of the 3 Pythagorean Trig Relationships

Examples: Simplifying Expressions � �

Part 3: Cofunction Trig Relationships �Sine & 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 Expressions �Simplify the following �tan (90° – A) = �Cos (π/2 – x) =

Tips to help simplify expressions �There are 4 different categories of trig relationships which each have different key components to look for �Reciprocal 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 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, 12, 13, 19