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# Hprec6 5

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### Hprec6 5

1. 1. 6-5: Basic Trigonometric Identities © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Develop the basic trig identities •Simplify trig expressions
2. 2. Definition A trigonometric identity is a statement of equality between two expressions. It means one expression can be used in place of the other. A list of the basic identities can be found on p.460 of your text.
3. 3. Example θ 4 3 5 5 csc 4 θ = 4 sin 5 θ = 1 5 4 = 1 cscθ =
4. 4. Definition 1 sin csc θ θ = 1 csc sin θ θ = 1 cos sec θ θ = 1 sec cos θ θ = 1 cot tan θ θ = 1 tan cot θ θ = Reciprocal Identities:
5. 5. Example If , find 1 sin 2 A = csc A
6. 6. Try This If , find 3 cos 4 β = sec β 4 sec 3 β =
7. 7. 4 3 5 Example 4sin 5 θ = 3cos 5 θ = 4 sin 5 3cos 5 θ θ = 4 45 tan 3 3 5 θ= = θ
8. 8. Definition cos cot sin A A A = sin tan A A cosA = sin cos tanA A A⇒ = Quotient Identities: cos sin cotA A A⇒ =
9. 9. Example If & 1 sin 2 A = tan A 2 cos 3 A = find
10. 10. Try This If & 1 cos 2 A = tan 2A = find sin A sin 1A =
11. 11. -1 1 -1 1 θ x y 1 sin 1 y yθ = = cos 1 x xθ = = but… 2 2 1x y+ = therefore 2 2 sin cos 1θ θ+ = Example
12. 12. Definition 2 2 sin cos 1θ θ+ = Divide by to get: 2 cos θ 2 2 tan 1 secθ θ+ = Pythagorean Identities:
13. 13. Definition 2 2 sin cos 1θ θ+ = Pythagorean Identities: Divide by to get: 2 sin θ 2 2 1 cot cscθ θ+ =
14. 14. Example Use the Pythagorean Identities to simplify the given expression: 2 2 2 tan cos cost t t+
15. 15. Example Use the Identities to simplify the given expression: 2 2 2 tan cos cost t t+
16. 16. Try This Use the Identities to simplify the given expression: 2 2 2 cot sin sint t t+ 1
17. 17. Try This Use the Identities to simplify the given expression : 2 2 2 sec tan cos t t t − 2 sec t
18. 18. Example Use the Pythagorean Identities to find sin t for the given value of cos t. Make sure the sign is correct for the given quadrant. 3 cos 10 t = − 2 t π π< <
19. 19. Try This Use the Pythagorean Identities to find sin t for the given value of cos t. Make sure the sign is correct for the given quadrant. 2 cos 5 t = 3 2 2 t π π< < 5 5 −
20. 20. Example If , find 3 tan 5 A = cos A first find …sec A
21. 21. Important Idea To solve trigonometric identity problems, you may use more than one identity in the same problem.
22. 22. Try This If , find 2 cos 3 θ = tanθ Assume θ is between 0 & 2 π 5 tan 2 θ =
23. 23. Example If and t is in quadrant I, find the 5 remaining trig functions. cos .3586t =
24. 24. Try This If and t is in quadrant II, find the 5 remaining trig functions. sin .2985t = cos .9544t = − tan .3128t = − sec 1.0478t = − csc 3.3501t = cot 3.1969t = −
25. 25. Example Simplify: 2 2 2 sin cos cos A A A +
26. 26. Try This Simplify: tan cscB B sec B
27. 27. Example Simplify: cos sec tan θ θ θ−
28. 28. Lesson Close From memory, name one trig identity we studied today.