Precalculus Examples

$\sin (\sec^{-1} (u))$

Step 1

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

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 $a = u$ and $i = 1$ .

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