Cosine of the difference and sum of two numbers

\displaystyle  \cos( A - B ) = \cos A \cos B + \sin A \sin B

\displaystyle  \cos( A + B ) = \cos A \cos B - \sin A \sin B

Sine of the difference and sum of two numbers

\displaystyle  \sin ( A - B ) = \sin A \cos B - \cos A \sin B

\displaystyle  \sin ( A + B ) = \sin A \cos B + \cos A \sin B

Tangent of the difference and sum of two numbers

\displaystyle  \tan ( A + B ) = \frac{\tan A + \tan B }{1 - \tan A \tan B}  

\displaystyle  \tan ( A - B ) = \frac{\tan A - \tan B }{1 + \tan A \tan B}  

Few more derivations based on the above formulas

\displaystyle  \sin ( A + B ) \sin ( A - B ) = \sin^2 A - \sin ^2 B = \cos^2 B - \cos^2 A

\displaystyle  \cos ( A + B ) \cos ( A - B ) = \cos^2 A - \sin ^2 B = \cos^2 B - \sin^2 A

\displaystyle  \sin ( A + B + C) = \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C

\displaystyle  \cos ( A + B + C) = \cos A \cos B \cos C - \cos A \sin B \sin C - \sin A \cos B \sin C - \sin A \sin B \cos C

\displaystyle  \tan ( A + B + C) = \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1 - \tan A \tan B - \tan B \tan C - \tan C \tan A }  

Formula to transform the product into sum or difference

\displaystyle  2 \sin A \cos B = \sin ( A + B ) + \sin ( A - B)

\displaystyle  2 \cos A \sin B = \sin ( A + B ) - \sin ( A - B)

\displaystyle  2 \cos A \cos B = \cos( A + B ) + \cos ( A - B)

\displaystyle  2 \sin A \sin B = \cos ( A - B ) - \cos ( A + B)

Formula to transform the sum or difference into product

\displaystyle  \sin C + \sin D = 2 \sin \Big( \frac{C+D}{2} \Big) \cos \Big( \frac{C-D}{2} \Big)

\displaystyle  \sin C - \sin D = 2 \sin \Big( \frac{C-D}{2} \Big) \cos \Big( \frac{C+D}{2} \Big)

\displaystyle  \cos C + \cos D = 2 \cos \Big( \frac{C+D}{2} \Big) \cos \Big( \frac{C-D}{2} \Big)

\displaystyle  \cos D - \cos C = 2 \sin \Big( \frac{C+D}{2} \Big) \sin \Big( \frac{C-D}{2} \Big)

\displaystyle  \cos C - \cos D = -2 \sin \Big( \frac{C+D}{2} \Big) \sin \Big( \frac{C-D}{2} \Big)

Value of trigonometric functions at \displaystyle  2x in terms of values at \displaystyle  x

\displaystyle  \sin 2 x = 2 \sin x \cos x

\displaystyle  \cos 2x = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x

\displaystyle  \tan 2x = \frac{2 \tan x }{1- \tan^2 x}  

\displaystyle  \sin 2x = \frac{2 \tan x }{1+ \tan^2 x}  

\displaystyle  \cos 2x = \frac{1 - \tan^2 x }{1+ \tan^2 x}  

Value of trigonometric functions at \displaystyle  3x in terms of values at \displaystyle  x

\displaystyle  \sin 3x = 3 \sin x - 4 \sin^3 x

\displaystyle  \cos 3x = 4 \cos^3 x - 3 \cos x

\displaystyle  \tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}  

Value of trigonometric functions at \displaystyle  x in terms of values at \displaystyle  \frac{x}{3}  

\displaystyle  \sin x = 3 \sin \frac{x}{3} - 4 \sin^3 \frac{x}{3}  

\displaystyle  \cos x = 4 \cos^3 \frac{x}{3} - 3 \cos \frac{x}{3}  

\displaystyle  \tan x = \frac{3 \tan \frac{x}{3} - \tan^3 \frac{x}{3}}{1 - 3 \tan^2 \frac{x}{3}}  

Values of trigonometrical functions at some important points

\displaystyle  \sin \frac{\pi}{10} = \frac{\sqrt{5}-1}{4}  

\displaystyle  \cos \frac{\pi}{10} = \frac{\sqrt{10 + 2 \sqrt{5}}}{4}  

\displaystyle  \cos \frac{\pi}{5} = \frac{\sqrt{5}+1}{4}  

\displaystyle  \sin \frac{\pi}{5} = \frac{\sqrt{10 - 2 \sqrt{5}}}{4}  

Law of Sines or Sine Rule

\displaystyle  \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k

Law of Cosines

\displaystyle  \cos A = \frac{b^2 + c^2 - a^2}{2 bc} or \displaystyle  a^2 = b^2 + c^2 - 2bc \cos A

\displaystyle  \cos B = \frac{c^2 + a^2 - b^2}{2 ca} or \displaystyle  b^2 = c^2 + a^2 - 2ca \cos B

\displaystyle  \cos C = \frac{a^2 + b^2 - c^2}{2 ab} or \displaystyle  c^2 = a^2 + b^2 - 2ab \cos A

Projection Formula

\displaystyle  a = b \cos C + c \cos B

\displaystyle  b = c \cos A + a \cos C

\displaystyle  c = a \cos B + b \cos A

Napier’s Analogy (law of tangents). In any \displaystyle  \triangle ABC , we have

\displaystyle  \tan \Big( \frac{B-C}{2} \Big) = \Big( \frac{b-c}{b+c} \Big) \cot \frac{A}{2}  

\displaystyle  \tan \Big( \frac{A-B}{2} \Big) = \Big( \frac{a-b}{a+b} \Big) \cot \frac{C}{2}  

\displaystyle  \tan \Big( \frac{C-A}{2} \Big) = \Big( \frac{c-a}{c+a} \Big) \cot \frac{B}{2}  

Area of a Triangle

\displaystyle  \triangle = \frac{1}{2} bc \sin A = \frac{1}{2} ca \sin B = \frac{1}{2} ab \sin C

Solving trigonometric equations

Step 1: Find a value of \displaystyle  x , preferably between \displaystyle  0 and \displaystyle  2\pi or between \displaystyle  -\pi and \displaystyle  \pi satisfying the given equation and call it \displaystyle  \alpha

Step 2:

  • If the equation is \displaystyle  \sin x = \sin \alpha , then \displaystyle  x = n\pi + ( -1)^n \alpha, n \in Z as a general solution
  • If the equation is \displaystyle  \cos x = \cos \alpha , then \displaystyle  x = 2n\pi \pm \alpha, n \in Z as a general solution
  • If the equation is \displaystyle  \tan x = \tan \alpha , then \displaystyle  x = n\pi + \alpha, n \in Z as a general solution