Question

Use a half-angle identity to find the exact value

Answer 1

Given:

`\cos 15^(\circ)`

To find:

The exact value of cos 15°.

Solution:

`$\cos 15^(\circ)=\cos( 30^(\circ))/(2)`

Using half-angle identity:

`$\cos \left((x)/(2)\right)=\sqrt{(1+\cos (x))/(2)}`

`$\cos (30^(\circ))/(2)=\sqrt{(1+\cos \left(30^(\circ)\right))/(2)}`

Using the trigonometric identity:
`\cos \left(30^(\circ)\right)=(√(3))/(2)`

`$=\sqrt{(1+(√(3))/(2))/(2)}`

Let us first solve the fraction in the numerator.

`$=\sqrt{((2+√(3))/(2))/(2)}`

Using fraction rule:
`((a)/(b) )/(c)=(a)/(b \cdot c)`

`$=\sqrt{\frac {2+√(3)}{4}}`

Apply radical rule:
`\sqrt[n]{(a)/(b)}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}`

`$=\frac{\sqrt{2+√(3)}}{√(4)}`

Using
`√(4) =2`:

`$=\frac{\sqrt{2+√(3)}}{2}`

`$\cos 15^\circ=\frac{\sqrt{2+√(3)}}{2}`