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Revision:Trigonometry - Addition Formulae

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Trigonometry: Addition Formulae

The results that must be learnt are:

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

Proof

It can be shown that

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

By considering sin(A - B) = sin(A + (-B)):

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

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

By considering cos(A + B) = sin(π/2 - (A + B)) = sin((π/2 - A) - B):

\cos(A+B) = \sin({\pi\over 2} - A) \cos B - \sin B \cos ({\pi\over 2} - A)

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

By considering cos(A - B) = cos(A + (-B)):

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

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

It is immediately obvious that

\displaystyle \tan(A+B) = \frac{\sin(A+B)}{\cos(A+B)} = \frac{\sin A\cos B + \sin B\cos A}{\cos A\cos B - \sin A\sin B} ,

which simplifies to the following identity when the numerator and denominator are divided by cos A and cos B:

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

Similarly,

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

We can derive the addition identities for cot in a similar way.

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

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

The addition identities for sec and csc are derived by taking the reciprocal of the identities for cos and sin, respectively, and multiplying the numerator and denominator by (sec A sec B csc A csc B).

\displaystyle \sec(A+B) = \frac{\sec A\sec B\csc A\csc B}{\csc A\csc B - \sec A\sec B}

\displaystyle \sec(A-B) = \frac{\sec A\sec B\csc A\csc B}{\csc A\csc B + \sec A\sec B}

\displaystyle \csc(A+B) = \frac{\sec A\sec B\csc A\csc B}{\sec A\csc B + \sec B\csc A}

\displaystyle \csc(A-B) = \frac{\sec A\sec B\csc A\csc B}{\sec A\csc B - \sec B\csc A}

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