Trigonometry Examples

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

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, $\sin (\arctan (\frac{x}{\sqrt{2}}))$ is $\frac{\frac{x}{\sqrt{2}}}{\sqrt{1 + {(\frac{x}{\sqrt{2}})}^{2}}}$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Combine fractions.

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Step 3.1

Combine.

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

Step 3.2

Multiply $x$ by $1$ .

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

Step 4

Simplify the denominator.

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Step 4.1

Combine using the product rule for radicals.

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

Step 4.2

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

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

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

Add $1$ and $1$ .

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

Step 4.3.6

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

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Step 4.3.6.1

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

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

Step 4.3.6.2

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

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

Step 4.3.6.3

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

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

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

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

Step 4.3.6.4.2

Rewrite the expression.

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

Step 4.3.6.5

Evaluate the exponent.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.5.4

Cancel the common factor of $2$ .

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Step 4.5.4.1

Cancel the common factor.

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

Step 4.5.4.2

Rewrite the expression.

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

Step 4.5.5

Evaluate the exponent.

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

Step 4.6

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

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

Cancel the common factors.

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Step 4.7.2.1

Factor $2$ out of $4$ .

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

Step 4.7.2.2

Cancel the common factor.

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

Step 4.7.2.3

Rewrite the expression.

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

Step 4.8

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

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

Step 4.9

Combine the numerators over the common denominator.

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

Reduce the expression by cancelling the common factors.

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Step 5.2.1

Reduce the expression $\frac{2 (2 + x^{2})}{2}$ by cancelling the common factors.

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Rewrite the expression.

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

Step 5.2.2

Divide $2 + x^{2}$ by $1$ .

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Simplify.

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