Precalculus Examples

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

Step 1

Subtract $\frac{3}{\sqrt{9 - x^{2}}}$ from both sides of the equation.

$\sec (\arcsin (\frac{x}{3})) - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2

Simplify $\sec (\arcsin (\frac{x}{3})) - \frac{3}{\sqrt{9 - x^{2}}}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{x}{3})}^{2}}, \frac{x}{3})$ , $(\sqrt{1^{2} - {(\frac{x}{3})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{x}{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{x}{3})}^{2}}, \frac{x}{3})$ . Therefore, $\sec (\arcsin (\frac{x}{3}))$ is $\frac{1}{\sqrt{1 - {(\frac{x}{3})}^{2}}}$ .

$\frac{1}{\sqrt{1 - {(\frac{x}{3})}^{2}}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

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

$\frac{1}{\sqrt{1^{2} - {(\frac{x}{3})}^{2}}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.2

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 $b = \frac{x}{3}$ .

$\frac{1}{\sqrt{(1 + \frac{x}{3}) (1 - \frac{x}{3})}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.3

Simplify.

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Step 2.1.2.3.1

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

$\frac{1}{\sqrt{(\frac{3}{3} + \frac{x}{3}) (1 - \frac{x}{3})}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.3.2

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{3 + x}{3} (1 - \frac{x}{3})}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.3.3

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

$\frac{1}{\sqrt{\frac{3 + x}{3} (\frac{3}{3} - \frac{x}{3})}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.3.4

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{3 + x}{3} \cdot \frac{3 - x}{3}}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.4

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

$\frac{1}{\sqrt{\frac{(3 + x) (3 - x)}{3 \cdot 3}}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.5

Multiply $3$ by $3$ .

$\frac{1}{\sqrt{\frac{(3 + x) (3 - x)}{9}}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.6

Rewrite $\frac{(3 + x) (3 - x)}{9}$ as ${(\frac{1}{3})}^{2} ((3 + x) (3 - x))$ .

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Step 2.1.2.6.1

Factor the perfect power $1^{2}$ out of $(3 + x) (3 - x)$ .

$\frac{1}{\sqrt{\frac{1^{2} ((3 + x) (3 - x))}{9}}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.6.2

Factor the perfect power $3^{2}$ out of $9$ .

$\frac{1}{\sqrt{\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.6.3

Rearrange the fraction $\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}$ .

$\frac{1}{\sqrt{{(\frac{1}{3})}^{2} ((3 + x) (3 - x))}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.7

Pull terms out from under the radical.

$\frac{1}{\frac{1}{3} \sqrt{(3 + x) (3 - x)}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.2.8

Combine $\frac{1}{3}$ and $\sqrt{(3 + x) (3 - x)}$ .

$\frac{1}{\frac{\sqrt{(3 + x) (3 - x)}}{3}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{3}{\sqrt{(3 + x) (3 - x)}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.4

Multiply $\frac{3}{\sqrt{(3 + x) (3 - x)}}$ by $1$ .

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

Step 2.1.5

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

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

Step 2.1.6

Combine and simplify the denominator.

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{{\sqrt{(3 + x) (3 - x)}}^{1} \sqrt{(3 + x) (3 - x)}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.6.3

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{{\sqrt{(3 + x) (3 - x)}}^{1} {\sqrt{(3 + x) (3 - x)}}^{1}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.6.4

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{{\sqrt{(3 + x) (3 - x)}}^{1 + 1}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.6.5

Add $1$ and $1$ .

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

Step 2.1.6.6

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

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Step 2.1.6.6.1

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{{({((3 + x) (3 - x))}^{\frac{1}{2}})}^{2}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.6.6.2

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{{((3 + x) (3 - x))}^{\frac{1}{2} \cdot 2}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.6.6.3

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

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

Step 2.1.6.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.6.4.1

Cancel the common factor.

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

Step 2.1.6.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{(3 + x) (3 - x)}}{{((3 + x) (3 - x))}^{1}} - \frac{3}{\sqrt{9 - x^{2}}} = 0$

Step 2.1.6.6.5

Simplify.

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

Step 2.1.7

Simplify the denominator.

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Step 2.1.7.1

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

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

Step 2.1.7.2

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - \frac{3}{\sqrt{(3 + x) (3 - x)}} = 0$

Step 2.1.8

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - (\frac{3}{\sqrt{(3 + x) (3 - x)}} \cdot \frac{\sqrt{(3 + x) (3 - x)}}{\sqrt{(3 + x) (3 - x)}}) = 0$

Step 2.1.9

Combine and simplify the denominator.

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Step 2.1.9.1

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - \frac{3 \sqrt{(3 + x) (3 - x)}}{\sqrt{(3 + x) (3 - x)} \sqrt{(3 + x) (3 - x)}} = 0$

Step 2.1.9.2

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - \frac{3 \sqrt{(3 + x) (3 - x)}}{{\sqrt{(3 + x) (3 - x)}}^{1} \sqrt{(3 + x) (3 - x)}} = 0$

Step 2.1.9.3

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - \frac{3 \sqrt{(3 + x) (3 - x)}}{{\sqrt{(3 + x) (3 - x)}}^{1} {\sqrt{(3 + x) (3 - x)}}^{1}} = 0$

Step 2.1.9.4

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - \frac{3 \sqrt{(3 + x) (3 - x)}}{{\sqrt{(3 + x) (3 - x)}}^{1 + 1}} = 0$

Step 2.1.9.5

Add $1$ and $1$ .

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

Step 2.1.9.6

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

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Step 2.1.9.6.1

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - \frac{3 \sqrt{(3 + x) (3 - x)}}{{({((3 + x) (3 - x))}^{\frac{1}{2}})}^{2}} = 0$

Step 2.1.9.6.2

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

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - \frac{3 \sqrt{(3 + x) (3 - x)}}{{((3 + x) (3 - x))}^{\frac{1}{2} \cdot 2}} = 0$

Step 2.1.9.6.3

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

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

Step 2.1.9.6.4

Cancel the common factor of $2$ .

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Step 2.1.9.6.4.1

Cancel the common factor.

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

Step 2.1.9.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - \frac{3 \sqrt{(3 + x) (3 - x)}}{{((3 + x) (3 - x))}^{1}} = 0$

Step 2.1.9.6.5

Simplify.

$\frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} - \frac{3 \sqrt{(3 + x) (3 - x)}}{(3 + x) (3 - x)} = 0$

Step 2.2

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

$0 = 0$