Trigonometry Examples

$\sin (\frac{19 π}{12})$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $\frac{19 π}{12}$ can be split into $\frac{5 π}{4} + \frac{π}{3}$ .

$\sin (\frac{5 π}{4} + \frac{π}{3})$

Step 2

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

$\sin (\frac{5 π}{4}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{4}) \sin (\frac{π}{3})$

Step 3

Remove parentheses.

$\sin (\frac{5 π}{4}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{4}) \sin (\frac{π}{3})$

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.

$- \sin (\frac{π}{4}) \cos (\frac{π}{3}) + \cos (\frac{5 π}{4}) \sin (\frac{π}{3})$

Step 4.2

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

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

Step 4.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \cos (\frac{5 π}{4}) \sin (\frac{π}{3})$

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Multiply $2$ by $2$ .

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

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.

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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Step 4.8.1

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

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

Step 4.8.2

Combine using the product rule for radicals.

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

Step 4.8.3

Multiply $3$ by $2$ .

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

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:

$- 0.96592582 \dots$