Trigonometry Examples

$\csc (\arccos (6 w))$

Step 1

Draw a triangle in the plane with vertices $(6 w, \sqrt{1^{2} - {(6 w)}^{2}})$ , $(6 w, 0)$ , and the origin. Then $\arccos (6 w)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(6 w, \sqrt{1^{2} - {(6 w)}^{2}})$ . Therefore, $\csc (\arccos (6 w))$ is $\frac{1}{\sqrt{1 - {(6 w)}^{2}}}$ .

$\frac{1}{\sqrt{1 - {(6 w)}^{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{1}{\sqrt{1^{2} - {(6 w)}^{2}}}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = 1$ and $i = 6 w$ .

$\frac{1}{\sqrt{(1 + 6 w) (1 - (6 w))}}$

Step 2.3

Multiply $6$ by $- 1$ .

$\frac{1}{\sqrt{(1 + 6 w) (1 - 6 w)}}$

Step 3

Multiply $\frac{1}{\sqrt{(1 + 6 w) (1 - 6 w)}}$ by $\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{\sqrt{(1 + 6 w) (1 - 6 w)}}$ .

$\frac{1}{\sqrt{(1 + 6 w) (1 - 6 w)}} \cdot \frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{\sqrt{(1 + 6 w) (1 - 6 w)}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{\sqrt{(1 + 6 w) (1 - 6 w)}}$ by $\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{\sqrt{(1 + 6 w) (1 - 6 w)}}$ .

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{\sqrt{(1 + 6 w) (1 - 6 w)} \sqrt{(1 + 6 w) (1 - 6 w)}}$

Step 4.2

Raise $\sqrt{(1 + 6 w) (1 - 6 w)}$ to the power of $1$ .

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{{\sqrt{(1 + 6 w) (1 - 6 w)}}^{1} \sqrt{(1 + 6 w) (1 - 6 w)}}$

Step 4.3

Raise $\sqrt{(1 + 6 w) (1 - 6 w)}$ to the power of $1$ .

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{{\sqrt{(1 + 6 w) (1 - 6 w)}}^{1} {\sqrt{(1 + 6 w) (1 - 6 w)}}^{1}}$

Step 4.4

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

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{{\sqrt{(1 + 6 w) (1 - 6 w)}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{{\sqrt{(1 + 6 w) (1 - 6 w)}}^{2}}$

Step 4.6

Rewrite ${\sqrt{(1 + 6 w) (1 - 6 w)}}^{2}$ as $(1 + 6 w) (1 - 6 w)$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + 6 w) (1 - 6 w)}$ as ${((1 + 6 w) (1 - 6 w))}^{\frac{1}{2}}$ .

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{{({((1 + 6 w) (1 - 6 w))}^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{{((1 + 6 w) (1 - 6 w))}^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{{((1 + 6 w) (1 - 6 w))}^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{{((1 + 6 w) (1 - 6 w))}^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{{((1 + 6 w) (1 - 6 w))}^{1}}$

Step 4.6.5

Simplify.

$\frac{\sqrt{(1 + 6 w) (1 - 6 w)}}{(1 + 6 w) (1 - 6 w)}$