Precalculus Examples

$\arccos (\frac{1}{\sqrt{3}} \cdot \sin (\frac{π}{3}))$

Step 1

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

$\arccos (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \cdot \sin (\frac{π}{3}))$

Step 2

Combine and simplify the denominator.

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Step 2.1

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

$\arccos (\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot \sin (\frac{π}{3}))$

Step 2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\arccos (\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot \sin (\frac{π}{3}))$

Step 2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\arccos (\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot \sin (\frac{π}{3}))$

Step 2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\arccos (\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot \sin (\frac{π}{3}))$

Step 2.5

Add $1$ and $1$ .

$\arccos (\frac{\sqrt{3}}{{\sqrt{3}}^{2}} \cdot \sin (\frac{π}{3}))$

Step 2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\arccos (\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot \sin (\frac{π}{3}))$

Step 2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\arccos (\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot \sin (\frac{π}{3}))$

Step 2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\arccos (\frac{\sqrt{3}}{3^{\frac{2}{2}}} \cdot \sin (\frac{π}{3}))$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\arccos (\frac{\sqrt{3}}{3^{\frac{2}{2}}} \cdot \sin (\frac{π}{3}))$

Step 2.6.4.2

Rewrite the expression.

$\arccos (\frac{\sqrt{3}}{3^{1}} \cdot \sin (\frac{π}{3}))$

Step 2.6.5

Evaluate the exponent.

$\arccos (\frac{\sqrt{3}}{3} \cdot \sin (\frac{π}{3}))$

Step 3

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

$\arccos (\frac{\sqrt{3}}{3} \cdot \frac{\sqrt{3}}{2})$

Step 4

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

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Step 4.1

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

$\arccos (\frac{\sqrt{3} \sqrt{3}}{3 \cdot 2})$

Step 4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\arccos (\frac{{\sqrt{3}}^{1} \sqrt{3}}{3 \cdot 2})$

Step 4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\arccos (\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{3 \cdot 2})$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\arccos (\frac{{\sqrt{3}}^{1 + 1}}{3 \cdot 2})$

Step 4.5

Add $1$ and $1$ .

$\arccos (\frac{{\sqrt{3}}^{2}}{3 \cdot 2})$

Step 4.6

Multiply $3$ by $2$ .

$\arccos (\frac{{\sqrt{3}}^{2}}{6})$

Step 5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\arccos (\frac{{(3^{\frac{1}{2}})}^{2}}{6})$

Step 5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\arccos (\frac{3^{\frac{1}{2} \cdot 2}}{6})$

Step 5.3

Combine $\frac{1}{2}$ and $2$ .

$\arccos (\frac{3^{\frac{2}{2}}}{6})$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$\arccos (\frac{3^{\frac{2}{2}}}{6})$

Step 5.4.2

Rewrite the expression.

$\arccos (\frac{3^{1}}{6})$

Step 5.5

Evaluate the exponent.

$\arccos (\frac{3}{6})$

Step 6

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

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Step 6.1

Factor $3$ out of $3$ .

$\arccos (\frac{3 (1)}{6})$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $3$ out of $6$ .

$\arccos (\frac{3 \cdot 1}{3 \cdot 2})$

Step 6.2.2

Cancel the common factor.

$\arccos (\frac{3 \cdot 1}{3 \cdot 2})$

Step 6.2.3

Rewrite the expression.

$\arccos (\frac{1}{2})$

Step 7

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

$\frac{π}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{3}$

Decimal Form:

$1.04719755 \dots$