Trigonometry Examples

$\sin (\arcsin (\frac{1}{2}) + \arccos (0))$

Step 1

Simplify each term.

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Step 1.1

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

$\sin (\frac{π}{6} + \arccos (0))$

Step 1.2

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$\sin (\frac{π}{6} + \frac{π}{2})$

Step 2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

Multiply $2$ by $3$ .

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

Step 4

Combine the numerators over the common denominator.

$\sin (\frac{π + π \cdot 3}{6})$

Step 5

Simplify the numerator.

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Step 5.1

Move $3$ to the left of $π$ .

$\sin (\frac{π + 3 \cdot π}{6})$

Step 5.2

Add $π$ and $3 π$ .

$\sin (\frac{4 π}{6})$

Step 6

Cancel the common factor of $4$ and $6$ .

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Step 6.1

Factor $2$ out of $4 π$ .

$\sin (\frac{2 (2 π)}{6})$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $6$ .

$\sin (\frac{2 (2 π)}{2 (3)})$

Step 6.2.2

Cancel the common factor.

$\sin (\frac{2 (2 π)}{2 \cdot 3})$

Step 6.2.3

Rewrite the expression.

$\sin (\frac{2 π}{3})$

Step 7

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin (\frac{π}{3})$

Step 8

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

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$