Expand Using Sum/Difference Formulas sin(-165)

$\sin (- 165)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $- 165$ can be split into $60 - 225$ .

$\sin (60 - 225)$

Step 2

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

$\sin (60) \cos (225) - \cos (60) \sin (225)$

Step 3

Remove parentheses.

$\sin (60) \cos (225) - \cos (60) \sin (225)$

Step 4

Simplify each term.

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

$\frac{\sqrt{3}}{2} \cos (225) - \cos (60) \sin (225)$

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

$\frac{\sqrt{3}}{2} (- \cos (45)) - \cos (60) \sin (225)$

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

$\frac{\sqrt{3}}{2} (- \frac{\sqrt{2}}{2}) - \cos (60) \sin (225)$

Multiply $\frac{\sqrt{3}}{2} (- \frac{\sqrt{2}}{2})$ .

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

$- \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \cos (60) \sin (225)$

Combine using the product rule for radicals.

$- \frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \cos (60) \sin (225)$

Multiply $3$ by $2$ .

$- \frac{\sqrt{6}}{2 \cdot 2} - \cos (60) \sin (225)$

Multiply $2$ by $2$ .

$- \frac{\sqrt{6}}{4} - \cos (60) \sin (225)$

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

$- \frac{\sqrt{6}}{4} - \frac{1}{2} \sin (225)$

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.

$- \frac{\sqrt{6}}{4} - \frac{1}{2} (- \sin (45))$

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

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

Multiply $- \frac{1}{2} (- \frac{\sqrt{2}}{2})$ .

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

Multiply $2$ by $2$ .

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

Step 5

Simplify.

Combine the numerators over the common denominator.

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

Factor $- 1$ out of $- \sqrt{6}$ .

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

Factor $- 1$ out of $\sqrt{2}$ .

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

Factor $- 1$ out of $- (\sqrt{6}) - 1 (- \sqrt{2})$ .

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

Simplify the expression.

Rewrite $- (\sqrt{6} - \sqrt{2})$ as $- 1 (\sqrt{6} - \sqrt{2})$ .

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

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.25881904 \dots$

Expand Using Sum/Difference Formulas sin(-165)

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