Trigonometry Examples

$\sin (arccot (3 x))$

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $3 x$ .

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

Step 2.2

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Rewrite ${\sqrt{9 x^{2} + 1}}^{2}$ as $9 x^{2} + 1$ .

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Simplify.

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