Expand Using Sum/Difference Formulas sin(105 degrees )

$\sin (105 °)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $105 °$ can be split into $45 + 60$ .

$\sin (45 + 60)$

Step 2

Use the sum formula for sine to simplify the expression. The formula states that $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

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

Step 3

Remove parentheses.

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

Step 4

Simplify each term.

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 $\frac{1}{2}$ .

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

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

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

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

Multiply $2$ by $2$ .

$\frac{\sqrt{2}}{4} + \cos (45) \sin (60)$

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

$\frac{\sqrt{2}}{4} + \frac{\sqrt{2}}{2} \sin (60)$

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

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

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

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

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

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 $2$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$

Expand Using Sum/Difference Formulas sin(105 degrees )

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