Trigonometry Examples

sin (\frac{17 π}{12})

$\sin (\frac{17 π}{12})$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case,

$\frac{17 π}{12}$ can be split into

$\frac{7 π}{6} + \frac{π}{4}$ .

$\sin (\frac{7 π}{6} + \frac{π}{4})$

Step 2

Use the sum formula for sine to simplify the expression. The formula states that

$\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

$\sin (\frac{7 π}{6}) \cos (\frac{π}{4}) + \cos (\frac{7 π}{6}) \sin (\frac{π}{4})$

Step 3

Remove parentheses.

$\sin (\frac{7 π}{6}) \cos (\frac{π}{4}) + \cos (\frac{7 π}{6}) \sin (\frac{π}{4})$

Step 4

Simplify each term.

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Step 4.1

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.

$- \sin (\frac{π}{6}) \cos (\frac{π}{4}) + \cos (\frac{7 π}{6}) \sin (\frac{π}{4})$

Step 4.2

The exact value of

$\sin (\frac{π}{6})$ is

$\frac{1}{2}$ .

$- \frac{1}{2} \cos (\frac{π}{4}) + \cos (\frac{7 π}{6}) \sin (\frac{π}{4})$

Step 4.3

The exact value of

$\cos (\frac{π}{4})$ is

$\frac{\sqrt{2}}{2}$ .

$- \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (\frac{7 π}{6}) \sin (\frac{π}{4})$

Step 4.4

Multiply

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

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Step 4.4.1

Multiply

$\frac{\sqrt{2}}{2}$ by

$\frac{1}{2}$ .

$- \frac{\sqrt{2}}{2 \cdot 2} + \cos (\frac{7 π}{6}) \sin (\frac{π}{4})$

Step 4.4.2

Multiply

$2$ by

$2$ .

$- \frac{\sqrt{2}}{4} + \cos (\frac{7 π}{6}) \sin (\frac{π}{4})$

Step 4.5

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.

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

Step 4.6

The exact value of

$\cos (\frac{π}{6})$ is

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

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

Step 4.7

The exact value of

$\sin (\frac{π}{4})$ is

$\frac{\sqrt{2}}{2}$ .

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

Step 4.8

Multiply

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

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Step 4.8.1

Multiply

$\frac{\sqrt{2}}{2}$ by

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

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

Step 4.8.2

Combine using the product rule for radicals.

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

Step 4.8.3

Multiply

$2$ by

$3$ .

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

Step 4.8.4

Multiply

$2$ by

$2$ .

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

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Factor

$- 1$ out of

$- \sqrt{2}$ .

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

Step 5.3

Factor

$- 1$ out of

$- \sqrt{6}$ .

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

Step 5.4

Factor

$- 1$ out of

$- (\sqrt{2}) - (\sqrt{6})$ .

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

Step 5.5

Simplify the expression.

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Step 5.5.1

Rewrite

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

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

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

Step 5.5.2

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.96592582 \dots$