Trigonometry Examples

$\sin (\arccos (\frac{\sqrt{2}}{2}) + \arctan (\sqrt{3}))$

Step 1

Simplify each term.

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Step 1.1

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

$\sin (\frac{π}{4} + \arctan (\sqrt{3}))$

Step 1.2

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

$\sin (\frac{π}{4} + \frac{π}{3})$

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $4$ by $3$ .

$\sin (\frac{π \cdot 3}{12} + \frac{π}{3} \cdot \frac{4}{4})$

Step 4.3

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

$\sin (\frac{π \cdot 3}{12} + \frac{π \cdot 4}{3 \cdot 4})$

Step 4.4

Multiply $3$ by $4$ .

$\sin (\frac{π \cdot 3}{12} + \frac{π \cdot 4}{12})$

Step 5

Combine the numerators over the common denominator.

$\sin (\frac{π \cdot 3 + π \cdot 4}{12})$

Step 6

Simplify the numerator.

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Step 6.1

Move $3$ to the left of $π$ .

$\sin (\frac{3 \cdot π + π \cdot 4}{12})$

Step 6.2

Move $4$ to the left of $π$ .

$\sin (\frac{3 π + 4 \cdot π}{12})$

Step 6.3

Add $3 π$ and $4 π$ .

$\sin (\frac{7 π}{12})$

Step 7

The exact value of $\sin (\frac{7 π}{12})$ is $\frac{\sqrt{2 + \sqrt{3}}}{2}$ .

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Step 7.1

Rewrite $\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\sin (\frac{\frac{7 π}{6}}{2})$

Step 7.2

Apply the sine half-angle identity.

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

Step 7.3

Change the $\pm$ to $+$ because sine is positive in the second quadrant.

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

Step 7.4

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

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Step 7.4.1

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 7.4.2

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

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

Step 7.4.3

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

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Step 7.4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 7.4.3.2

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

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

Step 7.4.4

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

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

Step 7.4.5

Combine the numerators over the common denominator.

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

Step 7.4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.4.7

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

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Step 7.4.7.1

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

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

Step 7.4.7.2

Multiply $2$ by $2$ .

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

Step 7.4.8

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

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

Step 7.4.9

Simplify the denominator.

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Step 7.4.9.1

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

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

Step 7.4.9.2

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$