28.4 Trig Identities & Equations Objectives:Identify the relationship of trig functions and positive and negative anglesIdentify the Pythagorean trig relationshipsIdentify the cofunction trig relationshipsApply various trig relationships to simplify expressions.Vocabulary: sine, cosine, tangent, cosecant, secant, cotangent, cofunction
3Review of Reciprocal Trig Relationships 𝑡𝑎𝑛θ= sin θ cos θ
4Example 1: Simplifying Expressions Simplify the following Expressions
5Part 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 = 1This is one of the PythagoreanTrig Relationships
7Part 1: Pythagorean Trig Relationships Starting with sin2θ + cos2θ = 1, how can you manipulate it to get other following Pythagorean Trig Relationships?1 + tan2θ = sec2θDivide both sides by cos2θ1 + cot2θ = csc2θDivide both sides by sin2θThese are the final 2 of the 3 Pythagorean Trig Relationships
9Part 2: Cofunction Trig Relationships Sine & Cosine, Tangent & Cotangent, Secant & Cosecant are all Cofunctions.Trig Cofunctions have the following relationshipWHY?
10Examples: Simplifying Expressions Simplify the followingtan (90° – A) =Cos (π/2 – x) =
11Part 3: Trig Relationships with Negative & Positive Angles Let’s take a look at a positive and negative angle on the unit circle
12Part 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?
13Part 3: Trig Relationships With Negative and Positive Angles We just proved that sin (-θ) = - sin θWhat do you think the relationship between cos (- θ) and cos θ is?cos (- θ) = cos θWhat about the relationship between tan (- θ) and tan θ?tan (- θ) = - tan θ
14Part 3: Trig Relationships With Negative and Positive Angles Let’s look at csc (- θ) and csc θ. What is the relationship?csc (- θ) = - csc θWhat about the relationship between sec (- θ) and sec θ?sec (- θ) = sec θWhat about the relationship between cot (- θ) and cot θ?cot (- θ) = - cot θ
15Examples: Practice Simplifying Write the equivalent trig function with a positive angleSin (-π/2)Cos (-π/3)Cot (-3π/4)
16Suggestions Change everything on both sides to sine and cosine. Start with the more complicated sideTry 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 fractionsIf you’re really stuck make sure to:Change everything on both sides to sine and cosine.Work with only one side at a time!
17Don’t Get Discouraged! Every identity is different Keep trying different approachesThe more you practice, the easier it will be to figure out efficient techniquesIf a solution eludes you at first, sleep on it! Try again the next day. Don’t give up!You will succeed!
18Tips to help simplify expressions There are 4 different categories of trig relationships which each have different key components to look forReciprocal RelationshipsMost commonly used in some type of format similar tocot y · sin ymanipulating a fraction with trig functionsUsually the functions aren’t squared when they are in this formatNegative/Positive Angle RelationshipsSimilar to the example problems previously in this powerpointtan (-45°)
19Tips to help simplify expressions There are 4 different categories of trig relationships which each have different key components to look forCofunction RelationshipsSimilar to the example problems previously in this powerpointcos (90° – A)Pythagorean Relationships (MOST COMMON/CHALLENGING!)Includes exponents to the second degreeIncludes expanding two binomialsAddition and subtraction of fractionsMay need to factor out a trig function before simplifyingOr some type of variation of the previous
20Tips 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