Trigonometry Examples

$\tan (\arcsin (\frac{5}{x}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{5}{x})}^{2}}, \frac{5}{x})$ , $(\sqrt{1^{2} - {(\frac{5}{x})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{5}{x})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{5}{x})}^{2}}, \frac{5}{x})$ . Therefore, $\tan (\arcsin (\frac{5}{x}))$ is $\frac{\frac{5}{x}}{\sqrt{1 - {(\frac{5}{x})}^{2}}}$ .

$\frac{\frac{5}{x}}{\sqrt{1 - {(\frac{5}{x})}^{2}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{5}{x} \cdot \frac{1}{\sqrt{1 - {(\frac{5}{x})}^{2}}}$

Step 3

Combine fractions.

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Step 3.1

Combine.

$\frac{5 \cdot 1}{x \sqrt{1 - {(\frac{5}{x})}^{2}}}$

Step 3.2

Multiply $5$ by $1$ .

$\frac{5}{x \sqrt{1 - {(\frac{5}{x})}^{2}}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $1$ as $1^{2}$ .

$\frac{5}{x \sqrt{1^{2} - {(\frac{5}{x})}^{2}}}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{5}{x}$ .

$\frac{5}{x \sqrt{(1 + \frac{5}{x}) (1 - \frac{5}{x})}}$

Step 4.3

Simplify.

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Step 4.3.1

Write $1$ as a fraction with a common denominator.

$\frac{5}{x \sqrt{(\frac{x}{x} + \frac{5}{x}) (1 - \frac{5}{x})}}$

Step 4.3.2

Combine the numerators over the common denominator.

$\frac{5}{x \sqrt{\frac{x + 5}{x} (1 - \frac{5}{x})}}$

Step 4.3.3

Write $1$ as a fraction with a common denominator.

$\frac{5}{x \sqrt{\frac{x + 5}{x} (\frac{x}{x} - \frac{5}{x})}}$

Step 4.3.4

Combine the numerators over the common denominator.

$\frac{5}{x \sqrt{\frac{x + 5}{x} \cdot \frac{x - 5}{x}}}$

Step 4.4

Multiply $\frac{x + 5}{x}$ by $\frac{x - 5}{x}$ .

$\frac{5}{x \sqrt{\frac{(x + 5) (x - 5)}{x \cdot x}}}$

Step 4.5

Multiply $x$ by $x$ .

$\frac{5}{x \sqrt{\frac{(x + 5) (x - 5)}{x^{2}}}}$

Step 4.6

Rewrite $\frac{(x + 5) (x - 5)}{x^{2}}$ as ${(\frac{1}{x})}^{2} ((x + 5) (x - 5))$ .

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Step 4.6.1

Factor the perfect power $1^{2}$ out of $(x + 5) (x - 5)$ .

$\frac{5}{x \sqrt{\frac{1^{2} ((x + 5) (x - 5))}{x^{2}}}}$

Step 4.6.2

Factor the perfect power $x^{2}$ out of $x^{2}$ .

$\frac{5}{x \sqrt{\frac{1^{2} ((x + 5) (x - 5))}{x^{2} \cdot 1}}}$

Step 4.6.3

Rearrange the fraction $\frac{1^{2} ((x + 5) (x - 5))}{x^{2} \cdot 1}$ .

$\frac{5}{x \sqrt{{(\frac{1}{x})}^{2} ((x + 5) (x - 5))}}$

Step 4.7

Pull terms out from under the radical.

$\frac{5}{x (\frac{1}{x} \sqrt{(x + 5) (x - 5)})}$

Step 4.8

Combine $\frac{1}{x}$ and $\sqrt{(x + 5) (x - 5)}$ .

$\frac{5}{x \frac{\sqrt{(x + 5) (x - 5)}}{x}}$

Step 5

Combine $x$ and $\frac{\sqrt{(x + 5) (x - 5)}}{x}$ .

$\frac{5}{\frac{x \sqrt{(x + 5) (x - 5)}}{x}}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Reduce the expression $\frac{x \sqrt{(x + 5) (x - 5)}}{x}$ by cancelling the common factors.

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Step 6.1.1

Cancel the common factor.

$\frac{5}{\frac{x \sqrt{(x + 5) (x - 5)}}{x}}$

Step 6.1.2

Rewrite the expression.

$\frac{5}{\frac{\sqrt{(x + 5) (x - 5)}}{1}}$

Step 6.2

Divide $\sqrt{(x + 5) (x - 5)}$ by $1$ .

$\frac{5}{\sqrt{(x + 5) (x - 5)}}$

Step 7

Multiply $\frac{5}{\sqrt{(x + 5) (x - 5)}}$ by $\frac{\sqrt{(x + 5) (x - 5)}}{\sqrt{(x + 5) (x - 5)}}$ .

$\frac{5}{\sqrt{(x + 5) (x - 5)}} \cdot \frac{\sqrt{(x + 5) (x - 5)}}{\sqrt{(x + 5) (x - 5)}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{5}{\sqrt{(x + 5) (x - 5)}}$ by $\frac{\sqrt{(x + 5) (x - 5)}}{\sqrt{(x + 5) (x - 5)}}$ .

$\frac{5 \sqrt{(x + 5) (x - 5)}}{\sqrt{(x + 5) (x - 5)} \sqrt{(x + 5) (x - 5)}}$

Step 8.2

Raise $\sqrt{(x + 5) (x - 5)}$ to the power of $1$ .

$\frac{5 \sqrt{(x + 5) (x - 5)}}{{\sqrt{(x + 5) (x - 5)}}^{1} \sqrt{(x + 5) (x - 5)}}$

Step 8.3

Raise $\sqrt{(x + 5) (x - 5)}$ to the power of $1$ .

$\frac{5 \sqrt{(x + 5) (x - 5)}}{{\sqrt{(x + 5) (x - 5)}}^{1} {\sqrt{(x + 5) (x - 5)}}^{1}}$

Step 8.4

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

$\frac{5 \sqrt{(x + 5) (x - 5)}}{{\sqrt{(x + 5) (x - 5)}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{5 \sqrt{(x + 5) (x - 5)}}{{\sqrt{(x + 5) (x - 5)}}^{2}}$

Step 8.6

Rewrite ${\sqrt{(x + 5) (x - 5)}}^{2}$ as $(x + 5) (x - 5)$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 5) (x - 5)}$ as ${((x + 5) (x - 5))}^{\frac{1}{2}}$ .

$\frac{5 \sqrt{(x + 5) (x - 5)}}{{({((x + 5) (x - 5))}^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{5 \sqrt{(x + 5) (x - 5)}}{{((x + 5) (x - 5))}^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{5 \sqrt{(x + 5) (x - 5)}}{{((x + 5) (x - 5))}^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{5 \sqrt{(x + 5) (x - 5)}}{{((x + 5) (x - 5))}^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{5 \sqrt{(x + 5) (x - 5)}}{{((x + 5) (x - 5))}^{1}}$

Step 8.6.5

Simplify.

$\frac{5 \sqrt{(x + 5) (x - 5)}}{(x + 5) (x - 5)}$