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7.1 – Basic Trigonometric Identities and Equations

7.1 – Basic Trigonometric Identities and Equations. Trigonometric Identities. Quotient Identities. Reciprocal Identities. Pythagorean Identities. sin 2 q + cos 2 q = 1. tan 2 q + 1 = sec 2 q. cot 2 q + 1 = csc 2 q. sin 2 q = 1 - cos 2 q. tan 2 q = sec 2 q - 1.

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7.1 – Basic Trigonometric Identities and Equations

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  1. 7.1 – Basic Trigonometric Identities and Equations

  2. Trigonometric Identities Quotient Identities Reciprocal Identities Pythagorean Identities sin2q + cos2q = 1 tan2q + 1 = sec2q cot2q + 1 = csc2q sin2q = 1 - cos2q tan2q = sec2q - 1 cot2q = csc2q - 1 cos2q = 1 - sin2q 5.4.3

  3. Where did our pythagorean identities come from?? Do you remember the Unit Circle? • What is the equation for the unit circle? x2 + y2 = 1 • What does x = ? What does y = ? • (in terms of trig functions) sin2θ + cos2θ = 1 Pythagorean Identity!

  4. Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos2θ sin2θ + cos2θ = 1 . cos2θcos2θ cos2θ tan2θ + 1 = sec2θ Quotient Identity Reciprocal Identity another Pythagorean Identity

  5. Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin2θ sin2θ + cos2θ = 1 . sin2θsin2θ sin2θ 1 + cot2θ = csc2θ Quotient Identity Reciprocal Identity a third Pythagorean Identity

  6. Using the identities you now know, find the trig value. 1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.

  7. 3.) sinθ = -1/3, find tanθ 4.) secθ = -7/5,find sinθ

  8. Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify. b) a) 5.4.5

  9. Simplifing Trigonometric Expressions c) (1 + tan x)2 - 2 sin x sec x d)

  10. Simplify each expression.

  11. sin x cos x Simplifying trig Identity Example1: simplify tanxcosx tanx cosx tanxcosx = sin x

  12. sec x sec x sin x sinx 1 1 1 = tan x = = x cos x cos x cos x csc x csc x sin x 1 Simplifying trig Identity Example2: simplify

  13. cos2x - sin2x cos2x - sin2x cos2x - sin2x 1 = sec x cos x cos x Simplifying trig Identity Example2: simplify

  14. Example Simplify: Factor out cot x = cot x (csc2 x - 1) Use pythagorean identity = cot x (cot2 x) Simplify = cot3 x

  15. = 1 = sin2 x + cos2x = sin x (sin x) + cos x = sin2 x + (cos x) cos x cos x cos x cos x cos x cos x Example Simplify: Use quotient identity Simplify fraction with LCD Simplify numerator Use pythagorean identity Use reciprocal identity = sec x

  16. Your Turn! Combine fraction Simplify the numerator Use pythagorean identity Use Reciprocal Identity

  17. Practice

  18. One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this: substitute using each identity simplify

  19. Another way to use identities is to write one function in terms of another function. Let’s see an example of this: This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

  20. (E) Examples • Prove tan(x) cos(x) = sin(x)

  21. (E) Examples • Prove tan2(x) = sin2(x) cos-2(x)

  22. (E) Examples • Prove

  23. (E) Examples • Prove

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