Sum and difference identities

Three quick questions from earlier lessons.

Is $\cos(30^\circ + 60^\circ)$ equal to $\cos 30^\circ + \cos 60^\circ$ ?

The sum and difference identities give exact values for angles like $75^\circ$ or $15^\circ$ .

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

.

\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B

(the sign flips for cosine).

Find the exact value of $\sin 75^\circ$ .

\sin 75^\circ = \sin 45^\circ\cos 30^\circ + \cos 45^\circ\sin 30^\circ

75° = 45° + 30°.

= \dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2}\cdot\dfrac{1}{2}

Special values.

= \dfrac{\sqrt{6} + \sqrt{2}}{4}

Common denominator.

Your turn: $\cos 15^\circ = \cos(45^\circ - 30^\circ)$ .

= \cos 45^\circ\cos 30^\circ + \sin 45^\circ\sin 30^\circ

For a difference, cosine takes a plus.

= \dfrac{\sqrt{6} + \sqrt{2}}{4}

√6/4 + √2/4.

Which expression is equal to $\cos(A - B)$ ?
1. $\cos A\cos B + \sin A\sin B$
2. $\cos A\cos B - \sin A\sin B$
3. $\cos A - \cos B$

$A$ and $B$ are in quadrant I, $\sin A = \frac{3}{5}$ and $\cos B = \dfrac{5}{13}$ . Find the exact value of $\sin(A + B)$ .
$A$ and $B$ are in quadrant I, $\sin A = \frac{7}{25}$ and $\cos B = \dfrac{5}{13}$ . Find the exact value of $\sin(A + B)$ .

Three questions to check you have got it.

Sum and difference identities

Watch out: $\cos(A + B) \ne \cos A + \cos B$ ; cosine flips the sign.

About this lesson