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Section 8.4: Trig Identities & Equations

Section 8.4: Trig Identities & Equations. Pre-Calculus. 8.4 Trig Identities & Equations. Objectives : Identify the relationship of trig functions and positive and negative angles Identify the Pythagorean trig relationships Identify the cofunction trig relationships

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Section 8.4: Trig Identities & Equations

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  1. Section 8.4: Trig Identities & Equations Pre-Calculus

  2. 8.4 Trig Identities & Equations Objectives: • Identify the relationship of trig functions and positive and negative angles • Identify the Pythagorean trig relationships • Identify the cofunction trig relationships • Apply various trig relationships to simplify expressions. Vocabulary:sine, cosine, tangent, cosecant, secant, cotangent, cofunction

  3. Review of Reciprocal Trig Relationships

  4. Example 1: Simplifying Expressions • Simplify the following Expressions

  5. 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 Pythagorean Trig Relationships

  6. Examples: Simplifying Expressions

  7. 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

  8. Examples: Simplifying Expressions

  9. Part 2:Cofunction Trig Relationships • Sine & Cosine, Tangent & Cotangent, Secant & Cosecant are all Cofunctions. • Trig Cofunctions have the following relationship • WHY?

  10. Examples: Simplifying Expressions • Simplify the following • tan (90° – A) = • Cos (π/2 – x) =

  11. Part 3:Trig Relationships with Negative & Positive Angles • Let’s take a look at a positive and negative angle on the unit circle

  12. 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?

  13. 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 cosθis? • cos (- θ) = cosθ • What about the relationship between tan (- θ) and tan θ? • tan (- θ) = - tan θ

  14. Part 3:Trig Relationships With Negative and Positive Angles • Let’s look at csc (- θ) and cscθ. What is the relationship? • csc (- θ) = - cscθ • 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

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

  16. 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!

  17. 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!

  18. 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°)

  19. 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

  20. 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

  21. Homework • Textbook pg 321: #1, 5, 13, 21, 31

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