Trigonometry Examples

$\tan (\cos^{-1} (8 x))$

Step 1

Draw a triangle in the plane with vertices $(8 x, \sqrt{1^{2} - {(8 x)}^{2}})$ , $(8 x, 0)$ , and the origin. Then $\cos^{-1} (8 x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(8 x, \sqrt{1^{2} - {(8 x)}^{2}})$ . Therefore, $\tan (\cos^{-1} (8 x))$ is $\frac{\sqrt{1 - {(8 x)}^{2}}}{8 x}$ .

$\frac{\sqrt{1 - {(8 x)}^{2}}}{8 x}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{\sqrt{1^{2} - {(8 x)}^{2}}}{8 x}$

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 $b = 8 x$ .

$\frac{\sqrt{(1 + 8 x) (1 - (8 x))}}{8 x}$

Step 2.3

Multiply $8$ by $- 1$ .

$\frac{\sqrt{(1 + 8 x) (1 - 8 x)}}{8 x}$