Find the Exact Value sin(-165 degrees )

$\sin (- 165 °)$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$- \sin (15)$

Step 2

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

$- \sin (45 - 30)$

Step 3

Separate negation.

$- \sin (45 - (30))$

Step 4

Apply the difference of angles identity.

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

Step 5

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

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

Step 6

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

Step 7

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

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

Step 8

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 9

Simplify $- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$ .

Simplify each term.

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

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

$- (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Combine using the product rule for radicals.

$- (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $2$ by $3$ .

$- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

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

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

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Combine the numerators over the common denominator.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.25881904 \dots$

Find the Exact Value sin(-165 degrees )

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