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.\(tan^{-1}x=cot^{-1}\frac{1}{x}\)
Trigonometry table
