Trigonometric Functions of Compound Angles

A compound angle is the one, which is formed by algebraic addition of 2 or more angles.

In algebraic addition, we consider the sign of the angle. Thus, \angle A and \angle B give \angle (A+B) and \angle A and - \angle B give \angle A + \ -B i.e. \angle (A-B).

Let A and B be any 2 angles.

  • Sum and Difference

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

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

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

***{cos (A-B) = cos(A)cos(B) \ + \ sin(A)sin(B)}

The formula *** is in some sense the base formula for all others. So, if we prove that this formula is true, rest of them can be proved very easily.

{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)}}

  • Allied Angles

Whenever the sum/ difference of any 2 angles is either zero or an integral multiple of \frac {\pi}{2}, they are known as allied angles. Using the sum and difference formulas and the ratios of standard angles, we can obtain the ratios of allied angles.

Let \theta be any angle. Then - \theta, \frac {\pi}{2} \pm \theta, \pi \pm \theta, \frac {3 \pi}{2} \pm \theta, 2 \pi - \theta are its allied angles.

allied_angles.jpg

 

  • Double Angle Formulas

{sin(2A) = 2sin(A)cos(A)}

{cos(2A) = cos^2(A)- sin^2 (A)}

{tan(2A) = \frac {2 tan(A)}{1- tan^2(A)}}

Recall, {sin^2(A)+cos^2(A) =1}.

Also, {sin (2A) = \frac {2 tan (A)}{1+ tan^2 (A)}}

{cos (2A) = \frac {1 - tan^2 (A)}{1+ tan^2 (A)}}

  • Triple Angle Formulas

{sin(3A)= 3sin(A)-4sin^3(A)}

{cos(3A)= 4cos^3(A)- 3cos (A)}

{tan(3A)= \frac {3tan(A)-tan^3(A)}{1-3tan^2(A)}}

  • Half Angle Formulas

Let \theta be any angle. Then,

{sin(\theta) = 2 sin (\frac \theta 2) cos (\frac \theta 2)}

{cos(\theta) = cos^2 (\frac \theta 2) - sin^2 (\frac \theta 2)}

{tan (\theta)= \frac {2 tan (\frac \theta 2)}{1 - tan^2 \frac \theta 2}}

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