Trigonometry Examples

$\sin (\arctan (\frac{x}{3}))$

Step 1

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

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Combine fractions.

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Step 3.1

Combine.

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

Step 3.2

Multiply $x$ by $1$ .

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

Step 4

Simplify the denominator.

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Step 4.1

Apply the product rule to $\frac{x}{3}$ .

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

Step 4.2

Raise $3$ to the power of $2$ .

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

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

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Step 4.5.1

Factor the perfect power $1^{2}$ out of $9 + x^{2}$ .

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

Step 4.5.2

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

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

Step 4.5.3

Rearrange the fraction $\frac{1^{2} (9 + x^{2})}{3^{2} \cdot 1}$ .

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

Step 4.6

Pull terms out from under the radical.

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

Step 4.7

Combine $\frac{1}{3}$ and $\sqrt{9 + x^{2}}$ .

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

Step 5

Combine $3$ and $\frac{\sqrt{9 + x^{2}}}{3}$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Reduce the expression $\frac{3 \sqrt{9 + x^{2}}}{3}$ by cancelling the common factors.

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Divide $\sqrt{9 + x^{2}}$ by $1$ .

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

Step 7

Multiply $\frac{x}{\sqrt{9 + x^{2}}}$ by $\frac{\sqrt{9 + x^{2}}}{\sqrt{9 + x^{2}}}$ .

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

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{x}{\sqrt{9 + x^{2}}}$ by $\frac{\sqrt{9 + x^{2}}}{\sqrt{9 + x^{2}}}$ .

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

Step 8.2

Raise $\sqrt{9 + x^{2}}$ to the power of $1$ .

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

Step 8.3

Raise $\sqrt{9 + x^{2}}$ to the power of $1$ .

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

Rewrite ${\sqrt{9 + x^{2}}}^{2}$ as $9 + x^{2}$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + x^{2}}$ as ${(9 + x^{2})}^{\frac{1}{2}}$ .

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Simplify.

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