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

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 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 2 θ + cos 2 θ = 1 This is one of the Pythagorean Trig Relationships

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