Trigonometry Examples

$x = \arcsin (\sqrt{\frac{32^{2}}{2 (32^{2}) + 64 (7)}})$

Step 1

Simplify $\arcsin (\sqrt{\frac{32^{2}}{2 (32^{2}) + 64 (7)}})$ .

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Step 1.1

Simplify the expression.

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Step 1.1.1

Raise $32$ to the power of $2$ .

$x = \arcsin (\sqrt{\frac{32^{2}}{2 \cdot 1024 + 64 (7)}})$

Step 1.1.2

Multiply $2$ by $1024$ .

$x = \arcsin (\sqrt{\frac{32^{2}}{2048 + 64 (7)}})$

Step 1.1.3

Multiply $64$ by $7$ .

$x = \arcsin (\sqrt{\frac{32^{2}}{2048 + 448}})$

Step 1.1.4

Add $2048$ and $448$ .

$x = \arcsin (\sqrt{\frac{32^{2}}{2496}})$

Step 1.1.5

Raise $32$ to the power of $2$ .

$x = \arcsin (\sqrt{\frac{1024}{2496}})$

Step 1.2

Cancel the common factor of $1024$ and $2496$ .

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Step 1.2.1

Factor $64$ out of $1024$ .

$x = \arcsin (\sqrt{\frac{64 (16)}{2496}})$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $64$ out of $2496$ .

$x = \arcsin (\sqrt{\frac{64 \cdot 16}{64 \cdot 39}})$

Step 1.2.2.2

Cancel the common factor.

$x = \arcsin (\sqrt{\frac{64 \cdot 16}{64 \cdot 39}})$

Step 1.2.2.3

Rewrite the expression.

$x = \arcsin (\sqrt{\frac{16}{39}})$

Step 1.3

Rewrite $\sqrt{\frac{16}{39}}$ as $\frac{\sqrt{16}}{\sqrt{39}}$ .

$x = \arcsin (\frac{\sqrt{16}}{\sqrt{39}})$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Rewrite $16$ as $4^{2}$ .

$x = \arcsin (\frac{\sqrt{4^{2}}}{\sqrt{39}})$

Step 1.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \arcsin (\frac{4}{\sqrt{39}})$

Step 1.5

Multiply $\frac{4}{\sqrt{39}}$ by $\frac{\sqrt{39}}{\sqrt{39}}$ .

$x = \arcsin (\frac{4}{\sqrt{39}} \cdot \frac{\sqrt{39}}{\sqrt{39}})$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

Multiply $\frac{4}{\sqrt{39}}$ by $\frac{\sqrt{39}}{\sqrt{39}}$ .

$x = \arcsin (\frac{4 \sqrt{39}}{\sqrt{39} \sqrt{39}})$

Step 1.6.2

Raise $\sqrt{39}$ to the power of $1$ .

$x = \arcsin (\frac{4 \sqrt{39}}{{\sqrt{39}}^{1} \sqrt{39}})$

Step 1.6.3

Raise $\sqrt{39}$ to the power of $1$ .

$x = \arcsin (\frac{4 \sqrt{39}}{{\sqrt{39}}^{1} {\sqrt{39}}^{1}})$

Step 1.6.4

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

$x = \arcsin (\frac{4 \sqrt{39}}{{\sqrt{39}}^{1 + 1}})$

Step 1.6.5

Add $1$ and $1$ .

$x = \arcsin (\frac{4 \sqrt{39}}{{\sqrt{39}}^{2}})$

Step 1.6.6

Rewrite ${\sqrt{39}}^{2}$ as $39$ .

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Step 1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{39}$ as $39^{\frac{1}{2}}$ .

$x = \arcsin (\frac{4 \sqrt{39}}{{(39^{\frac{1}{2}})}^{2}})$

Step 1.6.6.2

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

$x = \arcsin (\frac{4 \sqrt{39}}{39^{\frac{1}{2} \cdot 2}})$

Step 1.6.6.3

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

$x = \arcsin (\frac{4 \sqrt{39}}{39^{\frac{2}{2}}})$

Step 1.6.6.4

Cancel the common factor of $2$ .

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Step 1.6.6.4.1

Cancel the common factor.

$x = \arcsin (\frac{4 \sqrt{39}}{39^{\frac{2}{2}}})$

Step 1.6.6.4.2

Rewrite the expression.

$x = \arcsin (\frac{4 \sqrt{39}}{39^{1}})$

Step 1.6.6.5

Evaluate the exponent.

$x = \arcsin (\frac{4 \sqrt{39}}{39})$

Step 1.7

Evaluate $\arcsin (\frac{4 \sqrt{39}}{39})$ .

$x = 0.69516559$

Step 2

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

$x \approx 0.69516559$

Step 3