Precalculus Examples

$\sin (- \frac{7 π}{12})$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $- \frac{7 π}{12}$ can be split into $\frac{π}{4} - \frac{5 π}{6}$ .

$\sin (\frac{π}{4} - \frac{5 π}{6})$

Step 2

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

$\sin (\frac{π}{4}) \cos (\frac{5 π}{6}) - \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 3

Remove parentheses.

$\sin (\frac{π}{4}) \cos (\frac{5 π}{6}) - \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cos (\frac{5 π}{6}) - \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 4.2

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 second quadrant.

$\frac{\sqrt{2}}{2} (- \cos (\frac{π}{6})) - \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 4.3

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{2}}{2} (- \frac{\sqrt{3}}{2}) - \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 4.4

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

Tap for more steps...

Step 4.4.1

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

$- \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 4.4.2

Combine using the product rule for radicals.

$- \frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 4.4.3

Multiply $2$ by $3$ .

$- \frac{\sqrt{6}}{2 \cdot 2} - \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 4.4.4

Multiply $2$ by $2$ .

$- \frac{\sqrt{6}}{4} - \cos (\frac{π}{4}) \sin (\frac{5 π}{6})$

Step 4.5

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \sin (\frac{5 π}{6})$

Step 4.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 4.7

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 4.8

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

Tap for more steps...

Step 4.8.1

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

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

Step 4.8.2

Multiply $2$ by $2$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Simplify the expression.

Tap for more steps...

Step 5.5.1

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

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

Step 5.5.2

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:

$- 0.96592582 \dots$