Hprec6 5
- 1. 6-5: Basic Trigonometric Identities © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Develop the basic trig identities •Simplify trig expressions
- 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.
- 5. Example If , find 1 sin 2 A = csc A
- 6. Try This If , find 3 cos 4 β = sec β 4 sec 3 β =
- 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. 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. Example If & 1 sin 2 A = tan A 2 cos 3 A = find
- 10. Try This If & 1 cos 2 A = tan 2A = find sin A sin 1A =
- 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. Definition 2 2 sin cos 1θ θ+ = Divide by to get: 2 cos θ 2 2 tan 1 secθ θ+ = Pythagorean Identities:
- 13. Definition 2 2 sin cos 1θ θ+ = Pythagorean Identities: Divide by to get: 2 sin θ 2 2 1 cot cscθ θ+ =
- 14. Example Use the Pythagorean Identities to simplify the given expression: 2 2 2 tan cos cost t t+
- 15. Example Use the Identities to simplify the given expression: 2 2 2 tan cos cost t t+
- 16. Try This Use the Identities to simplify the given expression: 2 2 2 cot sin sint t t+ 1
- 17. Try This Use the Identities to simplify the given expression : 2 2 2 sec tan cos t t t − 2 sec t
- 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. 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. Example If , find 3 tan 5 A = cos A first find …sec A
- 21. Important Idea To solve trigonometric identity problems, you may use more than one identity in the same problem.
- 22. Try This If , find 2 cos 3 θ = tanθ Assume θ is between 0 & 2 π 5 tan 2 θ =
- 23. Example If and t is in quadrant I, find the 5 remaining trig functions. cos .3586t =
- 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. Example Simplify: 2 2 2 sin cos cos A A A +
- 26. Try This Simplify: tan cscB B sec B
- 28. Lesson Close From memory, name one trig identity we studied today.