Trigonometry Examples

$\csc (\arctan (\frac{x}{\sqrt{2}}))$

Step 1

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

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Multiply $\frac{x}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{x}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Evaluate the exponent.

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

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $\frac{x \sqrt{2}}{2}$ .

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

Step 5.2

Apply the product rule to $x \sqrt{2}$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

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

Step 6.5

Evaluate the exponent.

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

Step 7

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

$\sqrt{1 + \frac{x^{2} \cdot 2}{4}} \frac{\sqrt{2}}{x}$

Step 8

Cancel the common factor of $2$ and $4$ .

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Step 8.1

Factor $2$ out of $x^{2} \cdot 2$ .

$\sqrt{1 + \frac{2 \cdot x^{2}}{4}} \frac{\sqrt{2}}{x}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $4$ .

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

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

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

Step 12

Combine.

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

Step 13

Cancel the common factor of $\sqrt{2}$ .

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Step 13.1

Cancel the common factor.

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

Step 13.2

Rewrite the expression.

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