Precalculus Examples

$\cos (\arccos (0) - \arcsin (\frac{1}{2}))$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

$\cos (\frac{π}{2} - \frac{π}{6})$

Step 2

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

$\cos (\frac{π}{2} \cdot \frac{3}{3} - \frac{π}{6})$

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 $\frac{3}{3}$ .

$\cos (\frac{π \cdot 3}{2 \cdot 3} - \frac{π}{6})$

Step 3.2

Multiply $2$ by $3$ .

$\cos (\frac{π \cdot 3}{6} - \frac{π}{6})$

Step 4

Combine the numerators over the common denominator.

$\cos (\frac{π \cdot 3 - π}{6})$

Step 5

Simplify the numerator.

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Step 5.1

Move $3$ to the left of $π$ .

$\cos (\frac{3 \cdot π - π}{6})$

Step 5.2

Subtract $π$ from $3 π$ .

$\cos (\frac{2 π}{6})$

Step 6

Cancel the common factor of $2$ and $6$ .

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Step 6.1

Factor $2$ out of $2 π$ .

$\cos (\frac{2 (π)}{6})$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $6$ .

$\cos (\frac{2 π}{2 \cdot 3})$

Step 6.2.2

Cancel the common factor.

$\cos (\frac{2 π}{2 \cdot 3})$

Step 6.2.3

Rewrite the expression.

$\cos (\frac{π}{3})$

Step 7

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

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$