Precalculus Examples

$\cos (\arcsin (\frac{x - h}{r}))$

Step 1

Write the expression using exponents.

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Step 1.1

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

$\sqrt{1 - {(\frac{x - h}{r})}^{2}}$

Step 1.2

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

$\sqrt{1^{2} - {(\frac{x - h}{r})}^{2}}$

Step 2

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 $b = \frac{x - h}{r}$ .

$\sqrt{(1 + \frac{x - h}{r}) (1 - \frac{x - h}{r})}$

Step 3

Simplify.

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Step 3.1

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

$\sqrt{(\frac{r}{r} + \frac{x - h}{r}) (1 - \frac{x - h}{r})}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{\frac{r + x - h}{r} (1 - \frac{x - h}{r})}$

Step 3.3

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

$\sqrt{\frac{r + x - h}{r} (\frac{r}{r} - \frac{x - h}{r})}$

Step 3.4

Combine the numerators over the common denominator.

$\sqrt{\frac{r + x - h}{r} \cdot \frac{r - (x - h)}{r}}$

Step 3.5

Rewrite $\frac{r - (x - h)}{r}$ in a factored form.

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Step 3.5.1

Apply the distributive property.

$\sqrt{\frac{r + x - h}{r} \cdot \frac{r - x - - h}{r}}$

Step 3.5.2

Multiply $- - h$ .

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Step 3.5.2.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{r + x - h}{r} \cdot \frac{r - x + 1 h}{r}}$

Step 3.5.2.2

Multiply $h$ by $1$ .

$\sqrt{\frac{r + x - h}{r} \cdot \frac{r - x + h}{r}}$

Step 4

Multiply $\frac{r + x - h}{r}$ by $\frac{r - x + h}{r}$ .

$\sqrt{\frac{(r + x - h) (r - x + h)}{r \cdot r}}$

Step 5

Multiply $r$ by $r$ .

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

Step 6

Rewrite $\frac{(r + x - h) (r - x + h)}{r^{2}}$ as ${(\frac{1}{r})}^{2} ((r + x - h) (r - x + h))$ .

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Step 6.1

Factor the perfect power $1^{2}$ out of $(r + x - h) (r - x + h)$ .

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

Step 6.2

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

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

Step 6.3

Rearrange the fraction $\frac{1^{2} ((r + x - h) (r - x + h))}{r^{2} \cdot 1}$ .

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

Step 7

Pull terms out from under the radical.

$\frac{1}{r} \sqrt{(r + x - h) (r - x + h)}$

Step 8

Combine $\frac{1}{r}$ and $\sqrt{(r + x - h) (r - x + h)}$ .

$\frac{\sqrt{(r + x - h) (r - x + h)}}{r}$