Double angle formulae
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The following are important trigonometric relationships (it is unlikely that you will need to know how to prove them and they may be given in your formula book- check!):

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 -

sin(A - B) = sinAcosB - cosAsinB
cos(A - B) = cosAcosB + sinAsinB
tan(A - B) =   tanA - tanB
                    1 + tanAtanB

Double Angle Formulae
sin(A + B) = sinAcosB + cosAsinB
Replacing B by A in the above formula becomes:
sin(2A) = sinAcosA + cosAsinA
so sin2A = 2sinAcosA

similarly, cos2A = cosČA - sinČA
Replacing cosČA by 1 - sinČA (see Pythagorean identities) in the above formula gives:
cos2A = 1 - 2sinČA
Replacing sinČA by 1 - cosČA gives:
cos2A = 2cosČA - 1

It can also be shown that:
tan2A =    2tanA  
              1 - tanČA

© Matthew Pinkey

Other Notes in this Category

1. Double angle formulae
2. Pythagorean Identities
3. Radians
4. Sec, cosec, cot
5. Sin, Cos, Tan
6. Sine and Cosine Formulae
7. Solving Trigonometric Equations

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