Precalculus Examples

$\tan (\arccos (w))$

Step 1

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

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 $i = w$ .

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