Trigonometry Examples

$\csc (\arccos (u))$

Step 1

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

$\frac{1}{\sqrt{1 - u^{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{1}{\sqrt{1^{2} - u^{2}}}$

Step 2.2

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 $b = u$ .

$\frac{1}{\sqrt{(1 + u) (1 - u)}}$

Step 3

Multiply $\frac{1}{\sqrt{(1 + u) (1 - u)}}$ by $\frac{\sqrt{(1 + u) (1 - u)}}{\sqrt{(1 + u) (1 - u)}}$ .

$\frac{1}{\sqrt{(1 + u) (1 - u)}} \cdot \frac{\sqrt{(1 + u) (1 - u)}}{\sqrt{(1 + u) (1 - u)}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{\sqrt{(1 + u) (1 - u)}}$ by $\frac{\sqrt{(1 + u) (1 - u)}}{\sqrt{(1 + u) (1 - u)}}$ .

$\frac{\sqrt{(1 + u) (1 - u)}}{\sqrt{(1 + u) (1 - u)} \sqrt{(1 + u) (1 - u)}}$

Step 4.2

Raise $\sqrt{(1 + u) (1 - u)}$ to the power of $1$ .

$\frac{\sqrt{(1 + u) (1 - u)}}{{\sqrt{(1 + u) (1 - u)}}^{1} \sqrt{(1 + u) (1 - u)}}$

Step 4.3

Raise $\sqrt{(1 + u) (1 - u)}$ to the power of $1$ .

$\frac{\sqrt{(1 + u) (1 - u)}}{{\sqrt{(1 + u) (1 - u)}}^{1} {\sqrt{(1 + u) (1 - u)}}^{1}}$

Step 4.4

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

$\frac{\sqrt{(1 + u) (1 - u)}}{{\sqrt{(1 + u) (1 - u)}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Rewrite ${\sqrt{(1 + u) (1 - u)}}^{2}$ as $(1 + u) (1 - u)$ .

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

$\frac{\sqrt{(1 + u) (1 - u)}}{{((1 + u) (1 - u))}^{1}}$

Step 4.6.5

Simplify.

$\frac{\sqrt{(1 + u) (1 - u)}}{(1 + u) (1 - u)}$