Trigonometry Examples

$\sin (\frac{3 π}{4} + \frac{5 π}{6})$

Step 1

To write $\frac{3 π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sin (\frac{3 π}{4} \cdot \frac{3}{3} + \frac{5 π}{6})$

Step 2

To write $\frac{5 π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sin (\frac{3 π}{4} \cdot \frac{3}{3} + \frac{5 π}{6} \cdot \frac{2}{2})$

Step 3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3 π}{4}$ by $\frac{3}{3}$ .

$\sin (\frac{3 π \cdot 3}{4 \cdot 3} + \frac{5 π}{6} \cdot \frac{2}{2})$

Step 3.2

Multiply $4$ by $3$ .

$\sin (\frac{3 π \cdot 3}{12} + \frac{5 π}{6} \cdot \frac{2}{2})$

Step 3.3

Multiply $\frac{5 π}{6}$ by $\frac{2}{2}$ .

$\sin (\frac{3 π \cdot 3}{12} + \frac{5 π \cdot 2}{6 \cdot 2})$

Step 3.4

Multiply $6$ by $2$ .

$\sin (\frac{3 π \cdot 3}{12} + \frac{5 π \cdot 2}{12})$

Step 4

Combine the numerators over the common denominator.

$\sin (\frac{3 π \cdot 3 + 5 π \cdot 2}{12})$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $3$ by $3$ .

$\sin (\frac{9 π + 5 π \cdot 2}{12})$

Step 5.2

Multiply $2$ by $5$ .

$\sin (\frac{9 π + 10 π}{12})$

Step 5.3

Add $9 π$ and $10 π$ .

$\sin (\frac{19 π}{12})$

Step 6

The exact value of $\sin (\frac{19 π}{12})$ is $- \frac{\sqrt{2 + \sqrt{3}}}{2}$ .

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Step 6.1

Rewrite $\frac{19 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\sin (\frac{\frac{19 π}{6}}{2})$

Step 6.2

Apply the sine half-angle identity.

$\pm \sqrt{\frac{1 - \cos (\frac{19 π}{6})}{2}}$

Step 6.3

Change the $\pm$ to $-$ because sine is negative in the fourth quadrant.

$- \sqrt{\frac{1 - \cos (\frac{19 π}{6})}{2}}$

Step 6.4

Simplify $- \sqrt{\frac{1 - \cos (\frac{19 π}{6})}{2}}$ .

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Step 6.4.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}$

Step 6.4.2

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 third quadrant.

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

Step 6.4.3

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

$- \sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{2}}$

Step 6.4.4

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

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Step 6.4.4.1

Multiply $- 1$ by $- 1$ .

$- \sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{2}}$

Step 6.4.4.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$- \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}$

Step 6.4.5

Write $1$ as a fraction with a common denominator.

$- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{2}}$

Step 6.4.6

Combine the numerators over the common denominator.

$- \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}}$

Step 6.4.7

Multiply the numerator by the reciprocal of the denominator.

$- \sqrt{\frac{2 + \sqrt{3}}{2} \cdot \frac{1}{2}}$

Step 6.4.8

Multiply $\frac{2 + \sqrt{3}}{2} \cdot \frac{1}{2}$ .

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Step 6.4.8.1

Multiply $\frac{2 + \sqrt{3}}{2}$ by $\frac{1}{2}$ .

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

Step 6.4.8.2

Multiply $2$ by $2$ .

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

Step 6.4.9

Rewrite $\sqrt{\frac{2 + \sqrt{3}}{4}}$ as $\frac{\sqrt{2 + \sqrt{3}}}{\sqrt{4}}$ .

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

Step 6.4.10

Simplify the denominator.

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Step 6.4.10.1

Rewrite $4$ as $2^{2}$ .

$- \frac{\sqrt{2 + \sqrt{3}}}{\sqrt{2^{2}}}$

Step 6.4.10.2

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{\sqrt{2 + \sqrt{3}}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{2 + \sqrt{3}}}{2}$

Decimal Form:

$- 0.96592582 \dots$