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RideAnS [48]
3 years ago
7

Use a half-angle identity to find the exact value

Mathematics
1 answer:
Tatiana [17]3 years ago
5 0

Given:

\cos 15^{\circ}

To find:

The exact value of cos 15°.

Solution:

$\cos 15^{\circ}=\cos\frac{ 30^{\circ}}{2}

Using half-angle identity:

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

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

Using the trigonometric identity: \cos \left(30^{\circ}\right)=\frac{\sqrt{3}}{2}

            $=\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}

Let us first solve the fraction in the numerator.

            $=\sqrt{\frac{\frac{2+\sqrt{3}}{2}}{2}}

Using fraction rule: \frac{\frac{a}{b} }{c}=\frac{a}{b \cdot c}

            $=\sqrt{\frac {2+\sqrt{3}}{4}}

Apply radical rule: \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}

           $=\frac{\sqrt{2+\sqrt{3}}}{\sqrt{4}}

Using \sqrt{4} =2:

           $=\frac{\sqrt{2+\sqrt{3}}}{2}

$\cos 15^\circ=\frac{\sqrt{2+\sqrt{3}}}{2}

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