Trigonometry Examples

$x = \arcsin (\frac{- 18}{\sqrt{468}})$

Step 1

Simplify $\arcsin (\frac{- 18}{\sqrt{468}})$ .

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $468$ as $6^{2} \cdot 13$ .

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Step 1.1.1.1

Factor $36$ out of $468$ .

$x = \arcsin (\frac{- 18}{\sqrt{36 (13)}})$

Step 1.1.1.2

Rewrite $36$ as $6^{2}$ .

$x = \arcsin (\frac{- 18}{\sqrt{6^{2} \cdot 13}})$

Step 1.1.2

Pull terms out from under the radical.

$x = \arcsin (\frac{- 18}{6 \sqrt{13}})$

Step 1.2

Reduce the expression by cancelling the common factors.

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Step 1.2.1

Cancel the common factor of $- 18$ and $6$ .

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Step 1.2.1.1

Factor $6$ out of $- 18$ .

$x = \arcsin (\frac{6 \cdot - 3}{6 \sqrt{13}})$

Step 1.2.1.2

Cancel the common factors.

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Step 1.2.1.2.1

Factor $6$ out of $6 \sqrt{13}$ .

$x = \arcsin (\frac{6 \cdot - 3}{6 (\sqrt{13})})$

Step 1.2.1.2.2

Cancel the common factor.

$x = \arcsin (\frac{6 \cdot - 3}{6 \sqrt{13}})$

Step 1.2.1.2.3

Rewrite the expression.

$x = \arcsin (\frac{- 3}{\sqrt{13}})$

Step 1.2.2

Move the negative in front of the fraction.

$x = \arcsin (- \frac{3}{\sqrt{13}})$

Step 1.3

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = \arcsin (- (\frac{3}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}))$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$x = \arcsin (- \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}})$

Step 1.4.2

Raise $\sqrt{13}$ to the power of $1$ .

$x = \arcsin (- \frac{3 \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}})$

Step 1.4.3

Raise $\sqrt{13}$ to the power of $1$ .

$x = \arcsin (- \frac{3 \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}})$

Step 1.4.4

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

$x = \arcsin (- \frac{3 \sqrt{13}}{{\sqrt{13}}^{1 + 1}})$

Step 1.4.5

Add $1$ and $1$ .

$x = \arcsin (- \frac{3 \sqrt{13}}{{\sqrt{13}}^{2}})$

Step 1.4.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$x = \arcsin (- \frac{3 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}})$

Step 1.4.6.2

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

$x = \arcsin (- \frac{3 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}})$

Step 1.4.6.3

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

$x = \arcsin (- \frac{3 \sqrt{13}}{13^{\frac{2}{2}}})$

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

$x = \arcsin (- \frac{3 \sqrt{13}}{13^{\frac{2}{2}}})$

Step 1.4.6.4.2

Rewrite the expression.

$x = \arcsin (- \frac{3 \sqrt{13}}{13^{1}})$

Step 1.4.6.5

Evaluate the exponent.

$x = \arcsin (- \frac{3 \sqrt{13}}{13})$

Step 1.5

Evaluate $\arcsin (- \frac{3 \sqrt{13}}{13})$ .

$x = - 0.98279372$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.98279372$

Step 3