Trigonometry Examples

$\sin (\arccos (\frac{\sqrt{3}}{2}) + \arctan (1))$

Step 1

Simplify each term.

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Step 1.1

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

$\sin (\frac{π}{6} + \arctan (1))$

Step 1.2

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$\sin (\frac{π}{6} + \frac{π}{4})$

Step 2

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

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

Step 3

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

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

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 $\frac{π}{6}$ by $\frac{2}{2}$ .

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

Step 4.2

Multiply $6$ by $2$ .

$\sin (\frac{π \cdot 2}{12} + \frac{π}{4} \cdot \frac{3}{3})$

Step 4.3

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

$\sin (\frac{π \cdot 2}{12} + \frac{π \cdot 3}{4 \cdot 3})$

Step 4.4

Multiply $4$ by $3$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Move $2$ to the left of $π$ .

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

Step 6.2

Move $3$ to the left of $π$ .

$\sin (\frac{2 π + 3 \cdot π}{12})$

Step 6.3

Add $2 π$ and $3 π$ .

$\sin (\frac{5 π}{12})$

Step 7

The exact value of $\sin (\frac{5 π}{12})$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 7.1

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\sin (\frac{π}{6} + \frac{π}{4})$

Step 7.2

Apply the sum of angles identity.

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

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

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

Step 7.7

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

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Step 7.7.1

Simplify each term.

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Step 7.7.1.1

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

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Step 7.7.1.1.1

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

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

Step 7.7.1.1.2

Multiply $2$ by $2$ .

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

Step 7.7.1.2

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

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Step 7.7.1.2.1

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

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

Step 7.7.1.2.2

Combine using the product rule for radicals.

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

Step 7.7.1.2.3

Multiply $3$ by $2$ .

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

Step 7.7.1.2.4

Multiply $2$ by $2$ .

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

Step 7.7.2

Combine the numerators over the common denominator.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$