Find the Exact Value cos(15 degrees )

$\cos (15 °)$

Split $15 °$ into two angles where the values of the six trigonometric functions are known.

$\cos (45 - 30)$

Separate negation.

$\cos (45 - (30))$

Apply the difference of angles identity.

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

The exact value of $\cos (45)$ is $\frac{1}{\sqrt{2}}$ .

$\frac{1}{\sqrt{2}} \cos (30) + \sin (45) \sin (30)$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)$

The exact value of $\sin (45)$ is $\frac{1}{\sqrt{2}}$ .

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \sin (30)$

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

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

Simplify $\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2}$ .

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Simplify each term.

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Multiply $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{3}}{2}$ .

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

Move $2$ to the left of $\sqrt{2}$ .

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

Multiply $\frac{1}{\sqrt{2}}$ and $\frac{1}{2}$ .

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

Move $2$ to the left of $\sqrt{2}$ .

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

Combine the numerators over the common denominator.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$

Find the Exact Value cos(15 degrees )

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