Find the Exact Value sin(105)

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Trigonometry

$\sin (105)$

Split $105$ into two angles where the values of the six trigonometric functions are known.

$\sin (45 + 60)$

Apply the sum of angles identity.

$\sin (45) \cos (60) + \cos (45) \sin (60)$

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

$\frac{\sqrt{2}}{2} \cos (60) + \cos (45) \sin (60)$

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \cos (45) \sin (60)$

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

$\frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \frac{\sqrt{2}}{2} \sin (60)$

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

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

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

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Simplify each term.

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

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

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

Multiply $2$ by $2$ .

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

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}$

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2} + \sqrt{6}}{4}$

Decimal Form:

$0.96592582 \dots$

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