Trigonometry Review

1. Trigonometry Review Unit3 Identities

2. RECIPROCAL & QUOTIENT IDENTITIES r y x

3. PYTHAGOREAN IDENTITIES r y x cos2 q= 1 – sin 2q cos2 q+ sin 2q= 1 sin 2q + cos2 q= 1 sin 2q = 1 – cos2 q

4. PYTHAGOREAN IDENTITIES r y x 1 + tan 2q= sec2q tan 2q= sec2q -1

5. PYTHAGOREAN IDENTITIES r y x cot2q + 1= cos2 q 1 + cot2q= cos2 q cot2q= cos2 q - 1

6. 2 1 • For the following identities: • Show that it is true for • Prove the result algebraically

7. 2 1 • For the following identities: • Show that it is true for • Prove the result algebraically

8. SUM AND DIFFERENCE IDENTITIES sin (A + B) ≠ sin A + sin B sin (30o + 60o) ≠ sin 30o + sin 60o sin 90o ≠ sin 30o + sin 60o

9. SUM AND DIFFERENCE IDENTITIES sin (A + B) = sin A cos B+ sin B cos A sin (30o + 60o) = sin 30o cos 60o + sin 60ocos 30o

10. This could be shown for each of the sum and difference identities on your data sheet. sin(A + B) = sin A cos B + cos A sin B sin(A – B) = sin A cos B – cos A sin B cos (A + B) = cos A cos B – sin A sin B cos(A – B) = cos A cos B+ sin A sin B

11. DOUBLE ANGLE IDENTITIES sin(2x) = sin (x) + sin(x) = sin (x) cos (x) + sin (x) cos (x) = 2sin (x) cos (x) sin(A + B) = sin A cos B + sin B cos A

12. DOUBLE ANGLE IDENTITIES cos(A + B) = cos A cos B – sin B sin B cos (2x) = cos(x)cos (x) – sin(x)sin(x) = cos2 (x) – sin2 (x) cos2 (x) + sin2 (x) = 1 ALSO cos (2x) = 1 – sin2 (x) – sin2 (x) = 1 – 2sin2 (x) cos (2x) = cos2 (x) – (1 – cos2 (x)) = 2cos2(x) – 1

13. SUM AND DIFFERENCE IDENTITIES sin(x + p) = sin x cos p+ sin pcos x = sin x( – 1 ) + (0)cos x = – sin x sin(A + B) = sin A cos B + sin B cos A

14. SUM AND DIFFERENCE IDENTITIES sin(A – B) = sin A cos B – sin B cos A = (1)cos x – sin x (0) = cos x

15. SUM AND DIFFERENCE IDENTITIES cos(A – B) = cos A cos B + sin A sin B cos (x – p/2 ) = cos x cos p/2 + sin x sin p/2 = cos x (0) + sin x (1) = sin x

16. Simplify cos2x – sin2x – cos (2x) = cos2x – (1 – cos2x) – (2cos2x – 1) = cos2x – 1 + cos2x – 2cos2x + 1) = 0