Question

Use a suitable half-angle formula to find the exact value of cos(15°).

Answer 1

Remember that

`\cos ((x)/(2))=\pm\sqrt[]{\frac{1_{}+\cos x}{2}}`

For x=30 degrees

`\cos ((30^o)/(2))=\cos (15^o)=\sqrt[]{(1+\cos 30^o)/(2)}`

we know that

`\cos 30^o=\frac{\sqrt[]{3}}{2}`

substitute

`\cos (15^o)=\sqrt[]{\frac{1+\frac{\sqrt[]{3}}{2}}{2}}`
`\cos (15^o)=\sqrt[]{\frac{2+\sqrt[]{3}}{4}}`
`\cos (15^o)=\frac{\sqrt[]{2+\sqrt[]{3}}}{2}`