Evaluate sin(arctan(a/b))

$\sin (\arctan (\frac{a}{b}))$

Step 1

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

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Combine fractions.

Combine.

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

Multiply $a$ by $1$ .

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

Step 4

Simplify the denominator.

Apply the product rule to $\frac{a}{b}$ .

$\frac{a}{b \sqrt{1 + \frac{a^{2}}{b^{2}}}}$

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

$\frac{a}{b \sqrt{\frac{b^{2}}{b^{2}} + \frac{a^{2}}{b^{2}}}}$

Combine the numerators over the common denominator.

$\frac{a}{b \sqrt{\frac{b^{2} + a^{2}}{b^{2}}}}$

Rewrite $\frac{b^{2} + a^{2}}{b^{2}}$ as ${(\frac{1}{b})}^{2} (b^{2} + a^{2})$ .

Factor the perfect power $1^{2}$ out of $b^{2} + a^{2}$ .

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

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

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

Rearrange the fraction $\frac{1^{2} (b^{2} + a^{2})}{b^{2} \cdot 1}$ .

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

Pull terms out from under the radical.

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

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

$\frac{a}{b \frac{\sqrt{b^{2} + a^{2}}}{b}}$

Step 5

Combine $b$ and $\frac{\sqrt{b^{2} + a^{2}}}{b}$ .

$\frac{a}{\frac{b \sqrt{b^{2} + a^{2}}}{b}}$

Step 6

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{b \sqrt{b^{2} + a^{2}}}{b}$ by cancelling the common factors.

Cancel the common factor.

$\frac{a}{\frac{b \sqrt{b^{2} + a^{2}}}{b}}$

Rewrite the expression.

$\frac{a}{\frac{\sqrt{b^{2} + a^{2}}}{1}}$

Divide $\sqrt{b^{2} + a^{2}}$ by $1$ .

$\frac{a}{\sqrt{b^{2} + a^{2}}}$

Step 7

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

$\frac{a}{\sqrt{b^{2} + a^{2}}} \cdot \frac{\sqrt{b^{2} + a^{2}}}{\sqrt{b^{2} + a^{2}}}$

Step 8

Combine and simplify the denominator.

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

$\frac{a \sqrt{b^{2} + a^{2}}}{\sqrt{b^{2} + a^{2}} \sqrt{b^{2} + a^{2}}}$

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

$\frac{a \sqrt{b^{2} + a^{2}}}{{\sqrt{b^{2} + a^{2}}}^{1} \sqrt{b^{2} + a^{2}}}$

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

$\frac{a \sqrt{b^{2} + a^{2}}}{{\sqrt{b^{2} + a^{2}}}^{1} {\sqrt{b^{2} + a^{2}}}^{1}}$

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

$\frac{a \sqrt{b^{2} + a^{2}}}{{\sqrt{b^{2} + a^{2}}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{a \sqrt{b^{2} + a^{2}}}{{\sqrt{b^{2} + a^{2}}}^{2}}$

Rewrite ${\sqrt{b^{2} + a^{2}}}^{2}$ as $b^{2} + a^{2}$ .

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

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

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

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

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

$\frac{a \sqrt{b^{2} + a^{2}}}{{(b^{2} + a^{2})}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{a \sqrt{b^{2} + a^{2}}}{{(b^{2} + a^{2})}^{\frac{2}{2}}}$

Rewrite the expression.

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

Simplify.

$\frac{a \sqrt{b^{2} + a^{2}}}{b^{2} + a^{2}}$

Evaluate sin(arctan(a/b))

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