Trigonometry Examples

$\cot (\arcsin (\frac{a}{b}))$

Step 1

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

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Rewrite $1$ as $1^{2}$ .

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

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{a}{b}$ .

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

Step 5

Simplify terms.

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Step 5.1

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

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

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

$\sqrt{\frac{b + a}{b} (\frac{b}{b} - \frac{a}{b})} \frac{b}{a}$

Step 5.4

Combine the numerators over the common denominator.

$\sqrt{\frac{b + a}{b} \cdot \frac{b - a}{b}} \frac{b}{a}$

Step 5.5

Multiply $\frac{b + a}{b}$ by $\frac{b - a}{b}$ .

$\sqrt{\frac{(b + a) (b - a)}{b \cdot b}} \frac{b}{a}$

Step 5.6

Multiply $b$ by $b$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Pull terms out from under the radical.

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

Step 8

Combine $\frac{1}{b}$ and $\sqrt{(b + a) (b - a)}$ .

$\frac{\sqrt{(b + a) (b - a)}}{b} \cdot \frac{b}{a}$

Step 9

Cancel the common factor of $b$ .

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Step 9.1

Cancel the common factor.

$\frac{\sqrt{(b + a) (b - a)}}{b} \cdot \frac{b}{a}$

Step 9.2

Rewrite the expression.

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

Step 10

Combine $\sqrt{(b + a) (b - a)}$ and $\frac{1}{a}$ .

$\frac{\sqrt{(b + a) (b - a)}}{a}$