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# Trigonometric Ratios and Identities

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### Trigonometric Ratios and Identities

1. 1. TRIGONOMETRIC RATIOS AND IDENTITIES
2. 2. BASIC TRIGONOMETRIC IDENTITIES sin² θ + cos² θ = 1; -1 ≤ sin θ ≤ 1; -1 ≤ cos θ ≤ 1 θ R sec² θ - tan² θ = 1 ; sec θ ≥ 1 θ R– 2n 1 ,n 2 cosec² θ - cot² θ = 1 ; cosec θ ≥ 1 θ R – {n , n
3. 3. EXAMPLES
4. 4. CIRCULAR DEFINITION OF TRIGONOMETRICFUNCTIONS PM OM sin θ = cos θ = OP OP sin tan θ = , cos θ ≠ 0 cos cot θ = cos , sin θ ≠ 0 sin sec θ = 1 , cos θ ≠ 0 cos cosec θ = 1 , sin θ ≠ 0 sin
5. 5. TRIGONOMETRIC FUNCTIONS OF ALLIEDANGLES If θ is any angle, then - θ, 90 ± θ, 180 ± θ, 270 ± θ, 360 ± θ etc. are called Allied Angles. sin (- θ) = - sin θ ; cos (- θ) = cos θ sin (90° - θ) = cos θ ; cos (90° - θ) = sin θ sin (90° + θ) = cos θ ; cos (90° + θ) = - sin θ sin (180° - θ) = sin θ ; cos (180° - θ) = - cos θ sin (180° + θ) = - sin θ ; cos (180° + θ) = - cos θ sin (270° - θ) = - cos θ ; cos (270° - θ) = - sin θ sin (270° + θ) = - cos θ ; cos (270° + θ) = sin θ tan (90° - θ) = cot θ ; cot (90° - θ) = tan θ
6. 6. EXAMPLES Prove that (i) cot A + tan (180º + A) + tan (90º + A) + tan (360º – A) = 0 (ii) sec (270º – A) sec (90º – A) – tan (270º – A) tan (90º + A) + 1 = 0 Prove that (i) sin 420º cos 390º + cos (–300º) sin (–330º) = 1 (ii) tan 225º cot 405º + tan 765º cot 675º = 0
7. 7. GRAPHS OF TRIGONOMETRIC FUNCTIONS y = sin x x R; y [–1, 1]
8. 8.  y = cos x x R; y [ – 1, 1]
9. 9.  y = tan x x R – (2n + 1) /2, n I; y R
10. 10.  y = cot x x R–n ,n I; y R
11. 11.  y = cosec x x R –n ,n I; y (- ∞, - 1] [1, ∞)
12. 12.  y = sec x x R – (2n + 1) /2, n I; y (- ∞, - 1] [1, ∞)
13. 13. EXAMPLES
14. 14. TRIGONOMETRIC FUNCTIONS OF SUM ORDIFFERENCE OF TWO ANGLES  sin (A ± B) = sinA cosB ± cosA sinB  cos (A ± B) = cosA cosB  sinA sinB  sin²A - sin²B = cos²B - cos²A = sin (A+B). sin (A- B)  cos²A - sin²B = cos²B - sin²A = cos (A+B). cos (A - B)
15. 15. tan A tanB tan (A ± B) = 1  tan A tanB cot A cot B  1 cot (A ± B) = cot B cot A tan A tanB tanC tan A tanB tan C tan (A + B + C) = 1 tan A tanB tanB tan C tan C tan A
16. 16. EXAMPLES
17. 17. FACTORISATION OF THE SUM OR DIFFERENCE OFTWO SINES OR COSINES sinC + sinD = 2 sin C D cos C D 2 2 sinC - sinD = 2 cos C D sin C D 2 2 cosC + cosD = 2 cosC D cos D C 2 2 cosC - cosD = - 2 sinC D sin C D 2 2
18. 18. EXAMPLES
19. 19. TRANSFORMATION OF PRODUCTS INTO SUM ORDIFFERENCE OF SINES & COSINES 2 sinA cosB = sin(A+B) + sin(A-B) 2 cosA sinB = sin(A+B) - sin(A-B) 2 cosA cosB = cos(A+B) + cos(A-B) 2 sinA sinB = cos(A-B) - cos(A+B)
20. 20. EXAMPLES
21. 21. MULTIPLE AND SUB-MULTIPLE ANGLES sin 2A = 2 sinA cosA ; sin θ = 2 sin (θ/2) cos (θ/2) cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2 sin²A; 2 cos² (θ/2) = 1 + cos θ , 2 sin² = 1 - cos θ 2 tan A 2 tan 2 tan 2 A 2 ; tan 1 tan A 1 tan 2 2 2 tan A 1 tan 2 A sin 2 A ; cos 2 A 2 1 tan A 1 tan 2 A
22. 22.  sin 3A = 3 sinA - 4 sin3A cos 3A = 4 cos3A - 3 cosA 3 tan A tan3 A tan 3A = 1 3 tan2 A
23. 23. EXAMPLES
24. 24. IMPORTANT TRIGONOMETRIC RATIOS sin n = 0 ; cos n = (-1)n ; tan n = 0, where n I 3 1 5 sin 15° or sin = = cos 75° or cos ; 12 2 2 12 3 1 5 cos 15° or cos = = sin 75° or sin ; 12 2 2 12 3 1 tan 15° = = 2 3 = cot 75° ; 3 1 3 1 tan 75° = = 2 3 = cot 15° 3 1 5 1 5 1 sin or sin 18° = & cos 36° or cos = 10 4 5 4
25. 25. CONDITIONAL IDENTITIES If A + B + C = then : sin2A + sin2B + sin2C = 4 sinA sinB sinC sinA + sinB + sinC = 4 cos (A/2)cos(B/2) cos(C/2) cos 2 A + cos 2 B + cos 2 C = - 1 - 4 cos A cos B cos C cos A + cos B + cos C = 1 + 4 sin(A/2)sin(B/2) sin(C/2) tanA + tanB + tanC = tanA tanB tanC
26. 26. A B B C C A tan .tan tan .tan tan .tan 1 2 2 2 2 2 2 A B C A B C cot cot cot cot .cot .cot 2 2 2 2 2 2 cot A.cot B + cot B.cot C + cot C.cot A = 1 A + B + C = /2 then tan A.tan B + tan B.tan C + tan C.tan A = 1
27. 27. EXAMPLES
28. 28. RANGE OF TRIGONOMETRIC EXPRESSION E = a sin θ + b cos θ b E= a 2 b 2 sin (θ + ), where tan = a a2 b2 a = cos (θ - ), where tan = b Hence for any real value of θ, a2 b2 E a2 b2
29. 29. EXAMPLES
30. 30. SINE AND COSINE SERIES sin sin( ) sin( 2 ) .... sin n 1 sin n2 n 1 sin sin 2 2 cos cos( ) cos( 2 ) .... cos n 1 sin n2 n 1 cos sin 2 2
31. 31. EXAMPLES