# Trigonometric Formula

#### Some primary identities and rule for trigonometry:

1.$$Sin \theta = \frac{opposite side}{hypotenuse}$$

2.$$Cos \theta= \frac{adjacent side}{hypotenuse}$$

3. $$Tan θ = \frac{opposite side}{adjacent side}$$

4. $$Sec θ =\frac{ hypotenuse}{adjacent side}$$

5.$$Cosec θ = \frac{hypotenuse}{opposite side}$$

6.$$Cot θ = \frac{adjacent side}{opposite side}$$

7.$$sin^{2}\theta+cos^{2}\theta=1$$

8.$$sec^2\theta-tan^2\theta=1$$

9.$$cosec^2\theta-cot^2\theta=1$$

10.$$sin(-\theta)=-sin\theta$$

11.$$cosec(-\theta)=-cosec\theta$$

12.$$cos(-\theta)=cos\theta$$

13.$$sec(-\theta)=sec\theta$$

14.$$tan(-\theta)=-tan\theta$$

15.$$cot(-\theta)=-cot\theta$$

16.angel=$$(n\frac{π}{2}±\theta)$$;When n is even number› every trigonometric functions remine unchanged and when n is odd number every trigonometric functions will chenge with their cofunction like below-

sin>cos,

tan>cot,

sec>cosec,

cos>sin,

cot>tan &

cosec>sec .

#### Reciprocal Identities of trigonometry

17.$$Cosec θ = \frac{1}{sinθ}$$

18.$$Sec θ = \frac{1}{cosθ}$$

19.$$Cot θ = \frac{1}{tanθ}$$

20.$$Sin θ =\frac{1}{cosecθ }$$

21.$$Cos θ = \frac{1}{secθ}$$

22.$$sin\theta.cosec\theta=1$$

23.$$cos\theta.sec\theta=1$$

24.$$tan\theta.cot\theta=1$$

#### co-function trigonometric formula.

25.$$sin (\frac{π}{2} – A) = cos A$$

26.$$cos (\frac{π}{2} – A)= sin A$$

27.$$sin (\frac{π}{2} + A) = cos A$$

28.$$cos (\frac{π}{2} + A) = – sin A$$

29.$$sin (\frac{3π}{2} – A) = – cos A$$

30.$$cos (\frac{3π}{2}– A) = – sin A$$

31.$$sin (\frac{3π}{2} + A) = – cos A$$

32.$$cos (\frac{3π}{2} + A) = sin A$$

33.$$sin (π – A) = sin A$$

34.$$cos (π – A) = – cos A$$

35.$$sin(π + A) = – sin A$$

36.$$cos (π + A) = – cos A$$

37.$$sin (2π – A) = – sin A$$

38$$cos (2π – A) = cos A$$

39.$$sin (2π + A) = sin A$$

40.$$cos (2π + A) = cos A$$

41.$$sin(90°−A) = cos A$$

42.$$cos(90°−A) = sinA$$

43.$$tan(90°−A) = cot A$$

44.$$cot(90°−A) = tan A$$

45.$$sec(90°−A) = cosec A$$

46.$$cosec(90°−A) = sec A$$

#### Sum & Difference trigonometric formula:

47.$$sin(A+B) = sin(A)cos(B)+cos(A)sin(B)$$

48.$$cos(A+B) = cos(A)cos(B)–sin(A)sin(B)$$

49.$$tan(A+B) =\frac{ (tan A + tan B)}{(1−tan A tan B)}$$

50.$$cot(A+B)=\frac{cotA cotB-1}{cotB+cotA}$$

51.$$sin(A–B) = sin(A)cos(B)–cos(A)sin(B)$$

52.$$cos(A–B) = cos(A)cos(B) + sin(A)sin(B)$$

53.$$tan(A−B) =\frac{ (tan A–tan B)}{(1+tan A tan B)}$$

54.$$cot(A-B)=\frac{cotA cotB +1}{cotB-cotA}$$

#### Double Angle Identities

55.$$sin(2A) = 2sin(A) cos(A) = \frac{2tan A}{1+tan^{2} A}$$

56.$$cos(2A) = cos^{2}(A)–sin^{2}(A) = \frac{1-tan^2 A}{1+tan^2 A}$$

57.$$cos2A= 2cos^{2}(A)−1 = 1–2sin^{2}(A)$$

58.$$tan(2A) =\frac{2tan(A)}{1−tan^{2}(A)}$$

59.$$sec (2A) = \frac{sec^2 A}{2-sec^2 A}$$

60.$$cosec (2A) = \frac{sec A. cosec A}{2}$$

#### Triple Angle Identities

61.$$Sin 3A = 3sin A – 4sin^3A$$

62.$$Cos 3A = 4cos^3A-3cos A$$

63.$$Tan 3A= \frac{3tanx-tan^3A}{1-3tan^2A}$$

#### Half Angle Idntitiess

64.$$sin\frac{x}{2}=±\sqrt{\frac{1−cosx}{2}}$$

65.$$cos\frac{x}{2}=±\sqrt{\frac{1+cosx}{2}}$$

66.$$tan\frac{x}{2}=\sqrt{\frac{1−cosx}{1+cosx}}$$

Also,

67.$$tan\frac{x}{2}=\frac{1−cosx}{sinx}$$

#### Product identities

68.$$sinA⋅cosB=\frac{sin(A+B)+sin(A−B)}{2}$$

69.$$cosAsinB=\frac{sin(A+B)-sin(A-B)}{2}$$

70.$$cosA⋅cosB=\frac{cos(A+B)+cos(A−B)}{2}$$

71.$$sinA⋅sinB=\frac{cos(A−B)−cos(A+B)}{2}$$

#### Sum to Product Identities

72.$$sinC+sinD=2sin\frac{C+D}{2}cos\frac{C−D}{2}$$

73.$$sinC−sinD=2cos\frac{C+D}{2}sin\frac{C−D}{2}$$

74.$$cosC+cosD=2cos\frac{C+D}{2}cos\frac{C−D}{2}$$

75.$$cosC−coD=−2sin\frac{C+D}{2}sin\frac{C−D}{2}$$

#### Inverse Trigonometric Formulas

76.$$sin^{-1} (–x) = – sin^{-1 }x$$

77.$$cos^{-1} (–x) = π – cos^{-1} x$$

78.$$tan^{-1 }(–x) = – tan^{-1} x$$

79.$$cosec^{-1} (–x) = – cosec^{-1 }x$$

80.$$sec^{-1 }(–x) = π – sec^{-1} x$$

81.$$cot^{-1 }(–x) = π – cot^{-1 }x$$

Also,

82.$$sin^{-1}=cosec^{-1}\frac{1}{x}$$

83.$$cosec^{-1}x=sin^{-1}\frac{1}{x}$$

84.$$cos^{-1}x=sec^{-1}\frac{1}{x}$$

85.$$sec^{-1}x=cos^{-1}\frac{1}{x}$$

86.$$tan^{-1}x=cot^{-1}\frac{1}{x}$$

87.$$cot^{-1}x=tan^{-1}\frac{1}{x}$$
88.$$tan^{-1}x+tan^{-1}y=\frac{x+y}{1-xy}$$
89.$$tan^{-1}x-tan^{-1}=\frac{x-y}{1+xy}$$
90.$$2tan^{-1}x=\frac{2x}{1-x^2}$$