Trigonometry Examples

$\cot (\arcsin (\frac{\sqrt{x^{2} - 9}}{x}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{\sqrt{x^{2} - 9}}{x})}^{2}}, \frac{\sqrt{x^{2} - 9}}{x})$ , $(\sqrt{1^{2} - {(\frac{\sqrt{x^{2} - 9}}{x})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{\sqrt{x^{2} - 9}}{x})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{\sqrt{x^{2} - 9}}{x})}^{2}}, \frac{\sqrt{x^{2} - 9}}{x})$ . Therefore, $\cot (\arcsin (\frac{\sqrt{x^{2} - 9}}{x}))$ is $\frac{\sqrt{1 - {(\frac{\sqrt{x^{2} - 9}}{x})}^{2}}}{\frac{\sqrt{x^{2} - 9}}{x}}$ .

$\frac{\sqrt{1 - {(\frac{\sqrt{x^{2} - 9}}{x})}^{2}}}{\frac{\sqrt{x^{2} - 9}}{x}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{1 - {(\frac{\sqrt{x^{2} - 9}}{x})}^{2}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 3

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

$\sqrt{1^{2} - {(\frac{\sqrt{x^{2} - 9}}{x})}^{2}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 4

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 $b = \frac{\sqrt{x^{2} - 9}}{x}$ .

$\sqrt{(1 + \frac{\sqrt{x^{2} - 9}}{x}) (1 - \frac{\sqrt{x^{2} - 9}}{x})} \frac{x}{\sqrt{x^{2} - 9}}$

Step 5

Simplify.

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Step 5.1

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

$\sqrt{(\frac{x}{x} + \frac{\sqrt{x^{2} - 9}}{x}) (1 - \frac{\sqrt{x^{2} - 9}}{x})} \frac{x}{\sqrt{x^{2} - 9}}$

Step 5.2

Combine the numerators over the common denominator.

$\sqrt{\frac{x + \sqrt{x^{2} - 9}}{x} (1 - \frac{\sqrt{x^{2} - 9}}{x})} \frac{x}{\sqrt{x^{2} - 9}}$

Step 5.3

Rewrite $\frac{x + \sqrt{x^{2} - 9}}{x}$ in a factored form.

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Step 5.3.1

Rewrite $9$ as $3^{2}$ .

$\sqrt{\frac{x + \sqrt{x^{2} - 3^{2}}}{x} (1 - \frac{\sqrt{x^{2} - 9}}{x})} \frac{x}{\sqrt{x^{2} - 9}}$

Step 5.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\sqrt{\frac{x + \sqrt{(x + 3) (x - 3)}}{x} (1 - \frac{\sqrt{x^{2} - 9}}{x})} \frac{x}{\sqrt{x^{2} - 9}}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Rewrite $9$ as $3^{2}$ .

$\sqrt{\frac{x + \sqrt{(x + 3) (x - 3)}}{x} (1 - \frac{\sqrt{x^{2} - 3^{2}}}{x})} \frac{x}{\sqrt{x^{2} - 9}}$

Step 5.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\sqrt{\frac{x + \sqrt{(x + 3) (x - 3)}}{x} (1 - \frac{\sqrt{(x + 3) (x - 3)}}{x})} \frac{x}{\sqrt{x^{2} - 9}}$

Step 5.5

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

$\sqrt{\frac{x + \sqrt{(x + 3) (x - 3)}}{x} (\frac{x}{x} - \frac{\sqrt{(x + 3) (x - 3)}}{x})} \frac{x}{\sqrt{x^{2} - 9}}$

Step 5.6

Combine the numerators over the common denominator.

$\sqrt{\frac{x + \sqrt{(x + 3) (x - 3)}}{x} \cdot \frac{x - \sqrt{(x + 3) (x - 3)}}{x}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 6

Combine fractions.

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Step 6.1

Multiply $\frac{x + \sqrt{(x + 3) (x - 3)}}{x}$ by $\frac{x - \sqrt{(x + 3) (x - 3)}}{x}$ .

$\sqrt{\frac{(x + \sqrt{(x + 3) (x - 3)}) (x - \sqrt{(x + 3) (x - 3)})}{x \cdot x}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 6.2

Multiply $x$ by $x$ .

$\sqrt{\frac{(x + \sqrt{(x + 3) (x - 3)}) (x - \sqrt{(x + 3) (x - 3)})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 7

Expand $(x + \sqrt{(x + 3) (x - 3)}) (x - \sqrt{(x + 3) (x - 3)})$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$\sqrt{\frac{x (x - \sqrt{(x + 3) (x - 3)}) + \sqrt{(x + 3) (x - 3)} (x - \sqrt{(x + 3) (x - 3)})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 7.2

Apply the distributive property.

$\sqrt{\frac{x \cdot x + x (- \sqrt{(x + 3) (x - 3)}) + \sqrt{(x + 3) (x - 3)} (x - \sqrt{(x + 3) (x - 3)})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 7.3

Apply the distributive property.

$\sqrt{\frac{x \cdot x + x (- \sqrt{(x + 3) (x - 3)}) + \sqrt{(x + 3) (x - 3)} x + \sqrt{(x + 3) (x - 3)} (- \sqrt{(x + 3) (x - 3)})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8

Simplify terms.

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Step 8.1

Combine the opposite terms in $x \cdot x + x (- \sqrt{(x + 3) (x - 3)}) + \sqrt{(x + 3) (x - 3)} x + \sqrt{(x + 3) (x - 3)} (- \sqrt{(x + 3) (x - 3)})$ .

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Step 8.1.1

Reorder the factors in the terms $x (- \sqrt{(x + 3) (x - 3)})$ and $\sqrt{(x + 3) (x - 3)} x$ .

$\sqrt{\frac{x \cdot x - x \sqrt{(x + 3) (x - 3)} + x \sqrt{(x + 3) (x - 3)} + \sqrt{(x + 3) (x - 3)} (- \sqrt{(x + 3) (x - 3)})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.1.2

Add $- x \sqrt{(x + 3) (x - 3)}$ and $x \sqrt{(x + 3) (x - 3)}$ .

$\sqrt{\frac{x \cdot x + 0 + \sqrt{(x + 3) (x - 3)} (- \sqrt{(x + 3) (x - 3)})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.1.3

Add $x \cdot x$ and $0$ .

$\sqrt{\frac{x \cdot x + \sqrt{(x + 3) (x - 3)} (- \sqrt{(x + 3) (x - 3)})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2

Simplify each term.

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Step 8.2.1

Multiply $x$ by $x$ .

$\sqrt{\frac{x^{2} + \sqrt{(x + 3) (x - 3)} (- \sqrt{(x + 3) (x - 3)})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.2

Rewrite using the commutative property of multiplication.

$\sqrt{\frac{x^{2} - \sqrt{(x + 3) (x - 3)} \sqrt{(x + 3) (x - 3)}}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.3

Multiply $- \sqrt{(x + 3) (x - 3)} \sqrt{(x + 3) (x - 3)}$ .

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Step 8.2.3.1

Raise $\sqrt{(x + 3) (x - 3)}$ to the power of $1$ .

$\sqrt{\frac{x^{2} - ({\sqrt{(x + 3) (x - 3)}}^{1} \sqrt{(x + 3) (x - 3)})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.3.2

Raise $\sqrt{(x + 3) (x - 3)}$ to the power of $1$ .

$\sqrt{\frac{x^{2} - ({\sqrt{(x + 3) (x - 3)}}^{1} {\sqrt{(x + 3) (x - 3)}}^{1})}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.3.3

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

$\sqrt{\frac{x^{2} - {\sqrt{(x + 3) (x - 3)}}^{1 + 1}}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.3.4

Add $1$ and $1$ .

$\sqrt{\frac{x^{2} - {\sqrt{(x + 3) (x - 3)}}^{2}}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.4

Rewrite ${\sqrt{(x + 3) (x - 3)}}^{2}$ as $(x + 3) (x - 3)$ .

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Step 8.2.4.1

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

$\sqrt{\frac{x^{2} - {({((x + 3) (x - 3))}^{\frac{1}{2}})}^{2}}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.4.2

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

$\sqrt{\frac{x^{2} - {((x + 3) (x - 3))}^{\frac{1}{2} \cdot 2}}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.4.3

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

$\sqrt{\frac{x^{2} - {((x + 3) (x - 3))}^{\frac{2}{2}}}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.4.4

Cancel the common factor of $2$ .

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Step 8.2.4.4.1

Cancel the common factor.

$\sqrt{\frac{x^{2} - {((x + 3) (x - 3))}^{\frac{2}{2}}}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.4.4.2

Rewrite the expression.

$\sqrt{\frac{x^{2} - {((x + 3) (x - 3))}^{1}}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.4.5

Simplify.

$\sqrt{\frac{x^{2} - ((x + 3) (x - 3))}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.5

Expand $(x + 3) (x - 3)$ using the FOIL Method.

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Step 8.2.5.1

Apply the distributive property.

$\sqrt{\frac{x^{2} - (x (x - 3) + 3 (x - 3))}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.5.2

Apply the distributive property.

$\sqrt{\frac{x^{2} - (x \cdot x + x \cdot - 3 + 3 (x - 3))}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.5.3

Apply the distributive property.

$\sqrt{\frac{x^{2} - (x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3)}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.6

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 8.2.6.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

$\sqrt{\frac{x^{2} - (x \cdot x - 3 x + 3 x + 3 \cdot - 3)}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.6.2

Add $- 3 x$ and $3 x$ .

$\sqrt{\frac{x^{2} - (x \cdot x + 0 + 3 \cdot - 3)}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.6.3

Add $x \cdot x$ and $0$ .

$\sqrt{\frac{x^{2} - (x \cdot x + 3 \cdot - 3)}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.7

Simplify each term.

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Step 8.2.7.1

Multiply $x$ by $x$ .

$\sqrt{\frac{x^{2} - (x^{2} + 3 \cdot - 3)}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.7.2

Multiply $3$ by $- 3$ .

$\sqrt{\frac{x^{2} - (x^{2} - 9)}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.8

Apply the distributive property.

$\sqrt{\frac{x^{2} - x^{2} - - 9}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.2.9

Multiply $- 1$ by $- 9$ .

$\sqrt{\frac{x^{2} - x^{2} + 9}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.3

Simplify by adding terms.

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Step 8.3.1

Subtract $x^{2}$ from $x^{2}$ .

$\sqrt{\frac{0 + 9}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.3.2

Simplify the expression.

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Step 8.3.2.1

Add $0$ and $9$ .

$\sqrt{\frac{9}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 8.3.2.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{\frac{3^{2}}{x^{2}}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 9

Rewrite $\frac{3^{2}}{x^{2}}$ as ${(\frac{3}{x})}^{2}$ .

$\sqrt{{(\frac{3}{x})}^{2}} \frac{x}{\sqrt{x^{2} - 9}}$

Step 10

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

$\frac{3}{x} \cdot \frac{x}{\sqrt{x^{2} - 9}}$

Step 11

Simplify terms.

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Step 11.1

Combine.

$\frac{3 x}{x \sqrt{x^{2} - 9}}$

Step 11.2

Cancel the common factor of $x$ .

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Step 11.2.1

Cancel the common factor.

$\frac{3 x}{x \sqrt{x^{2} - 9}}$

Step 11.2.2

Rewrite the expression.

$\frac{3}{\sqrt{x^{2} - 9}}$

Step 12

Simplify the denominator.

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Step 12.1

Rewrite $9$ as $3^{2}$ .

$\frac{3}{\sqrt{x^{2} - 3^{2}}}$

Step 12.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{3}{\sqrt{(x + 3) (x - 3)}}$

Step 13

Multiply $\frac{3}{\sqrt{(x + 3) (x - 3)}}$ by $\frac{\sqrt{(x + 3) (x - 3)}}{\sqrt{(x + 3) (x - 3)}}$ .

$\frac{3}{\sqrt{(x + 3) (x - 3)}} \cdot \frac{\sqrt{(x + 3) (x - 3)}}{\sqrt{(x + 3) (x - 3)}}$

Step 14

Combine and simplify the denominator.

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Step 14.1

Multiply $\frac{3}{\sqrt{(x + 3) (x - 3)}}$ by $\frac{\sqrt{(x + 3) (x - 3)}}{\sqrt{(x + 3) (x - 3)}}$ .

$\frac{3 \sqrt{(x + 3) (x - 3)}}{\sqrt{(x + 3) (x - 3)} \sqrt{(x + 3) (x - 3)}}$

Step 14.2

Raise $\sqrt{(x + 3) (x - 3)}$ to the power of $1$ .

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

Step 14.3

Raise $\sqrt{(x + 3) (x - 3)}$ to the power of $1$ .

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

Step 14.4

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

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

Step 14.5

Add $1$ and $1$ .

$\frac{3 \sqrt{(x + 3) (x - 3)}}{{\sqrt{(x + 3) (x - 3)}}^{2}}$

Step 14.6

Rewrite ${\sqrt{(x + 3) (x - 3)}}^{2}$ as $(x + 3) (x - 3)$ .

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Step 14.6.1

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

$\frac{3 \sqrt{(x + 3) (x - 3)}}{{({((x + 3) (x - 3))}^{\frac{1}{2}})}^{2}}$

Step 14.6.2

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

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

Step 14.6.3

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

$\frac{3 \sqrt{(x + 3) (x - 3)}}{{((x + 3) (x - 3))}^{\frac{2}{2}}}$

Step 14.6.4

Cancel the common factor of $2$ .

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Step 14.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{(x + 3) (x - 3)}}{{((x + 3) (x - 3))}^{\frac{2}{2}}}$

Step 14.6.4.2

Rewrite the expression.

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

Step 14.6.5

Simplify.

$\frac{3 \sqrt{(x + 3) (x - 3)}}{(x + 3) (x - 3)}$