Cosine of the difference and sum of two numbers

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

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

Sine of the difference and sum of two numbers

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

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

Tangent of the difference and sum of two numbers

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

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

Few more derivations based on the above formulas

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

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

\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

\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

\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

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

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

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

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

Formula to transform the sum or difference into product

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

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

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

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

or

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\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

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

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

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

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

Law of Sines or Sine Rule

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

Law of Cosines

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

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

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

Projection Formula

a = b \cos C + c \cos B

b = c \cos A + a \cos C

c = a \cos B + b \cos A

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

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

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

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

Area of a Triangle

\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 x , preferably between 0 and 2\pi or between -\pi and \pi satisfying the given equation and call it \alpha

Step 2:

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