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5.1 Using the Fundamental Identities. 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.
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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