Precalculus Examples

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

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

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} ⎞ ⎠

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

⎛ ⎝ \sqrt{1^{2} - {(\frac{x - h}{r})}^{2}}, 0 ⎞ ⎠

$(\sqrt{1^{2} - {(\frac{x - h}{r})}^{2}}, 0)$ , and the origin. Then

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

$\arcsin (\frac{x - h}{r})$ 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} ⎞ ⎠

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

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

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

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

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

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

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

Step 1.2

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

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

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 1

$a = 1$ and

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

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

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

$1$ as a fraction with a common denominator.

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

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

Step 3.2

Combine the numerators over the common denominator.

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

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

Step 3.3

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 3.4

Combine the numerators over the common denominator.

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

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

Step 3.5

Rewrite

\frac{r - (x - h)}{r}

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

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

Step 3.5.2

Multiply

- - h

$- - h$ .

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Step 3.5.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 3.5.2.2

Multiply

h

$h$ by

1

$1$ .

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

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

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

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

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

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

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

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

Step 4

Multiply

\frac{r + x - h}{r}

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

\frac{r - x + h}{r}

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

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

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

Step 5

Multiply

r

$r$ by

r

$r$ .

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

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

Step 6

Rewrite

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

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

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

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

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Step 6.1

Factor the perfect power

1^{2}

$1^{2}$ out of

(r + x - h) (r - x + h)

$(r + x - h) (r - x + h)$ .

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

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

Step 6.2

Factor the perfect power

r^{2}

$r^{2}$ out of

r^{2}

$r^{2}$ .

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

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

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

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