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. 3. Example 4 3 5 5 csc 4 = 4 sin 5 = 1 5 4 = 1 csc = 4. Definition 1 sin csc = 1 csc sin = 1 cos sec = 1 sec cos = 1 cot tan = 1 tan cot = Reciprocal Identities: 5. Example If , find 1 sin 2 A = csc A 6. Try This If , find 3 cos 4 = sec 4 sec 3 = 7. 4 3 5 Example 4sin 5 = 3cos 5 = 4 sin 5 3cos 5 = 4 45 tan 3 3 5 = = 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 x y 1 sin 1 y y = = cos 1 x x = = but 2 2 1x y+ = therefore 2 2 sin cos 1 + = Example 12. Definition 2 2 sin cos 1 + = Divide by to get: 2 cos 2 2 tan 1 sec + = Pythagorean Identities: 13. Definition 2 2 sin cos 1 + = Pythagorean Identities: Divide by to get: 2 sin 2 2 1 cot csc + = 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 < < 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 < < 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 = tan Assume is between 0 & 2 5 tan 2 = 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 27. Example Simplify: cos sec tan 28. Lesson Close From memory, name one trig identity we studied today.