Trigonometry Examples

$\sin (2 \arctan (x))$

Step 1

Apply the sine double-angle identity.

$2 \sin (\arctan (x)) \cos (\arctan (x))$

Step 2

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

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

Step 3

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{x}{\sqrt{1 + x^{2}}}$ by $\frac{\sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Rewrite ${\sqrt{1 + x^{2}}}^{2}$ as $1 + x^{2}$ .

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Simplify.

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

Step 5

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

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

Step 6

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

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

Step 7

Cancel the common factor of $\sqrt{1 + x^{2}}$ .

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Step 7.1

Factor $\sqrt{1 + x^{2}}$ out of $2 x \sqrt{1 + x^{2}}$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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