Compound Angles Multiple Angles

The algebraic sum of two or more angles are generally called compound angles and the angles are known as the constituent angles.

sin(A + B) is not equal to sinA + sinB. Instead, you must expand such expressions using the formulae below.

The following are important trigonometric relationships:

sin(A + B) = sinAcosB + cosAsinB

cos(A + B) = cosAcosB - sinAsinB

tan(A + B) = tanA + tanB/1 - tanAtanB

To find sin(A - B), cos(A - B) and tan(A - B), just change the + signs in the above identities to - signs and vice-versa:

sin(A - B) = sinAcosB - cosAsinB

cos(A - B) = cosAcosB + sinAsinB

tan(A - B) = tanA - tanB/1 + tanAtanB

The trigonometric functions of the sum or difference of two angles can be expressed in terms of the functions of the individual angles.

Multiple Angles:

Sin(A + B) = sinAcosB + cosAsinB

Replacing B by A in the above formula becomes:

sin(2A) = sinAcosA + cosAsinA

so: sin2A = 2sinAcosA

similarly:

cos2A = cos2A - sin2A

Replacing cos2A by 1 - sin2A (using Pythagorean identities) in the above formula gives:

cos2A = 1 - 2sin2A

Replacing sin2A by 1 - cos2A gives:

cos2A = 2cos2A - 1

It can also be shown that:

tan2A = 2tanA/1 - tan2A

Transformation Formulae:

Let us establish two sets of transformation formulae: one is to transform the product of two sines or two cosines or one sine and one cosine into sum or difference of two sines or two cosines and the other to convert the sum or difference of two sines or two cosines in the product of two sines or two cosines or

one sine and one cosine.

2sinAcosB = sin(A+B)+sin(A-B)

2cosAsinB= sin(A+B) - sin(A-B)

2cosAcosB = cos(A+B) +cos(A-b)

2sinAsinB = cos(A-b) - cos(A+B)