Question

Find the exact value of cos15

Answer 1

Answer:

$\cos (15)=\frac{\sqrt{6}+\sqrt{2}}{4}$

Step-by-step explanation:

We want to find the exact value of

$\cos 15\degree$

We rewrite this expression using compound angles  

$\cos (45\degree-30\degree)$

Recall that:

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

We apply this property to obtain:

$\cos (45-30)=\cos 45\cos 30+\sin 45\sin 30$

$\cos (15)=\frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2} \times \frac{1}{2}$

$\cos (15)=\frac{\sqrt{6}+\sqrt{2}}{4}$

Answer 2

$\cos(15^\circ) = \cos(45^{\circ}-30^{\circ}) = \cos(45^{\circ})\cos(30^{\circ})+\sin(45^{\circ})\sin(30^{\circ}) = \\ \\ =\dfrac{\sqrt 2}{2}\cdot \dfrac{\sqrt 3}{2}+\dfrac{\sqrt 2}{2}\cdot \dfrac{1}{2} = \dfrac{\sqrt 6 +\sqrt 2}{4}$