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5.1 Using the Fundamental Identities

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## 5.1 Using the Fundamental Identities

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**4 Main Goals:**• Evaluate Trig Functions • Simplify Trig Expressions • Develop Additional Trig Identities • Solve Trig Equations**Fundamental Trig Identities (page 354)**Reciprocal Identities**Fundamental Trig Identities (page 354)**Quotient Identities**Fundamental Trig Identities (page 354)**Pythagorean Identities**Fundamental Trig Identities (page 354)**Cofunction Identities**Fundamental Trig Identities (page 354)**Even/Odd Identities**Using the identities**• Use the values and Tan θ > 0 to find the values of all 6 functions.**Using the identities**• Use the values and Tan θ > 0 to find the values of all 6 functions.**Using the identities**• Csc x = 4; Cos x < 0**Using the identities**• Csc x = 4; Cos x < 0**Using the Identities**• Tan θ is undefined, Sin θ > 0 Tan θ is undefined → Cos θ = 0**Using the Identities**• Tan θ is undefined, Sin θ > 0 Sin θ = 1 Cos θ = 0 Tan θ is undef. Sec θ = Cot θ = 0 Cscθ = undef. 1**Simplifying Trig Expressions**• To simplify a trig expression means to reduce it to simplest term • This typically means reducing a larger expression to 1 trig function • Never want any fractions in our answer (reciprocal identities)**Keep in mind:**• As we continue through the chapter, the problems with increase in difficulty • Always try to use the identities when possible • Last Resort is to convert all to sines and cosines • A common mistake is starting all problems by converting all to sines and cosines. Do this last!**Factoring**• So far, all the problems we have done have involved using the identities • Now, your first step should be to look to factor, then try to use the identities • What do you know how to factor?**Factoring**• Factor out a term Sin x Cos² - Sin x Sin x (Cos²x – 1) • Factor a trinomial Sin²x - 5Sin x + 6 (Sin x – 2) (Sin x – 3) • Factor special polynomials Sin³x - Sin²x – Sin x + 1 (Sin²x – 1) (Sin x - 1)**Simplify the following**• Sin x Cos²x – Sin x Can we factor? Sin x (Cos²x – 1) Sin x (Sin²x) Sin³ x**Simplify the following**If you get stuck, let x = Tan x 4x² + x - 3 = (2x + 3) (x – 1)**Trig Substitution**• Use the substitution x = 2 Tan θ to express the following expression as a trig function of θ**1. Substitute 2 Tan θ for x**2. Apply the rules for exponents 3. Factor 4. Simplify**Simplify the following:**• x = 3 Sin θ in the expression • x = 2 Tan θ in the expression • x = 2 Cos θ in the expression