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Trig Identities Quotient Identities sin tan cos cos cot sin Reciprocal Identities 1 sin csc cos 1 sec tan 1 cot OR 1 csc sin OR sec 1 cos OR cot 1 tan Pythagorean Identities sin 2 cos 2 1 OR sin 2 1 cos 2 1 tan 2 sec 2 OR tan 2 sec 2 1 cot 2 1 csc 2 OR cot 2 csc 2 1 OR cos 2 1 sin 2 Tips for proving trigonometric identities: 1. You want to make the left and right hand sides of the identities match by substitution and cancellation. 2. Work with the more complicated side of the identity. 3. Begin by writing all expressions in terms of sine and/or cosine. 4. If there is a squared term, check to see if you can use one of the Pythagorean identities. If so, use it to replace the squared term. 5. You are finished when the left hand side of the identity EXACTLY matches the right side. You can not move a term from one side to the other side. Handout Before we do some identities, lets practice substituting and cancelling. Write each expression as a single function or a constant. 1. 1 cos 2 Hint: look at trig identities! sin 2 3. tan cot Hint: change to sin and/or cos. sin cos 1 cos sin 5. tan csc 1 sin 1 cos sin cos cos sec 7. cos tan 2 1 sec 2 1 1 cos 2 cos cos sec Handout Write each expression as a single function or a constant. sin 2 2 tan 2 cos 1 9. 1 1 sec 2 cos 2 sin 2 cos 2 1 1 sin 2 cos 2 2 cos 1 Now we will try some with given ratios. 5 11. If cos and lies in Quadrant II, find 13 the values of the five remaining five trigonome tric functions. 12 a2 b2 c2 sin 52 b 2 132 13 5 25 b 2 169 cos 13 2 13 b 144 12 b 12 12 tan 5 5 13 12 13 sec 5 csc cot 5 12 Handout 4 and sin 0, find the values of the remaining 3 five trigonome tric functions. 7 csc sin 4 3 cos sec 2 2 2 4 a b c 32 b 2 4 2 7 cot tan 2 9 b 16 3 4 2 b 7 7 b 7 4 7 4 7 csc 3 7 7 7 13. If sec cot 3 7 3 7 7 7 7 4 7 4 3 3 7 Handout 5 15. If csc and the third quadrant, then cos 4 4 5 3 cos 5 sin 5 4 3 a2 b2 c2 4 2 b 2 52 16 b 2 25 b2 9 b3 csc 5 4 Handout 17. If sin .6 and cos 0, then tan sin 10 6 8 a2 b2 c2 6 2 b 2 10 2 36 b 2 100 b 2 64 b 8 6 10 tan 6 8 tan .75 Handout 19. If sin 3 and is an acute angle, what is the value of tan cos ? 4 3 4 7 cos 4 3 tan 7 sin 4 3 7 a2 b2 c2 32 b 2 4 2 9 b 2 16 b2 7 b 7 3 7 tan cos 7 4 3 4 Homework • Handout #2-20 evens 2. sin csc 1 sin 1 sin 8. sin 2 cot 2 cos 2 1 cot 2 csc 2 4. sec 2 1 6. tan 2 csc x sec x 1 sin x 1 cos x cos x 1 sin x 1 sin x cos x cot x 10. sin 2 cos tan csc sin 1 sin cos cos sin 2 sin 2 7 and cos 0, find 25 the values of the five remaining five trigonome tric functions. 7 a2 b2 c2 sin 2 2 2 7 b 25 25 24 49 b 2 625 cos 25 25 b 2 576 7 b 24 7 tan 24 24 12. If sin sin 2 2 tan 2 cos 14. sin 2 sin 2 1 sin 2 1 1 cos 2 sin 2 cos 2 sec 2 25 csc 7 25 sec 24 cot 24 7 16. sin 2 cos 2 tan 2 1 tan 2 18. cos k , then the value of cos sin cot sec 2 cos sin cos sin cos 2 k 2 20. The expression sec 2 csc 2 is equivalent to 1 1 2 2 cos sin 1 sin 2 1 cos 2 2 2 2 2 cos sin sin cos sin 2 cos 2 2 2 2 cos sin sin cos 2 sin 2 cos 2 cos 2 sin 2 1 cos 2 sin 2