Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

$\sin (\frac{π}{6} + \frac{π}{3})$

Step 2

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

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

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 $\frac{π}{3}$ by $\frac{2}{2}$ .

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

Step 3.2

Multiply $3$ by $2$ .

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

Step 4

Combine the numerators over the common denominator.

$\sin (\frac{π + π \cdot 2}{6})$

Step 5

Simplify the numerator.

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Step 5.1

Move $2$ to the left of $π$ .

$\sin (\frac{π + 2 \cdot π}{6})$

Step 5.2

Add $π$ and $2 π$ .

$\sin (\frac{3 π}{6})$

Step 6

Cancel the common factor of $3$ and $6$ .

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Step 6.1

Factor $3$ out of $3 π$ .

$\sin (\frac{3 (π)}{6})$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $3$ out of $6$ .

$\sin (\frac{3 π}{3 \cdot 2})$

Step 6.2.2

Cancel the common factor.

$\sin (\frac{3 π}{3 \cdot 2})$

Step 6.2.3

Rewrite the expression.

$\sin (\frac{π}{2})$

Step 7

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

$1$