Trigonometry Examples

$\cos (\arcsin (4 x))$

Step 1

Write the expression using exponents.

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Step 1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(4 x)}^{2}}, 4 x)$ , $(\sqrt{1^{2} - {(4 x)}^{2}}, 0)$ , and the origin. Then $\arcsin (4 x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(4 x)}^{2}}, 4 x)$ . Therefore, $\cos (\arcsin (4 x))$ is $\sqrt{1 - {(4 x)}^{2}}$ .

$\sqrt{1 - {(4 x)}^{2}}$

Step 1.2

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

$\sqrt{1^{2} - {(4 x)}^{2}}$

Step 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 $b = 4 x$ .

$\sqrt{(1 + 4 x) (1 - (4 x))}$

Step 3

Multiply $4$ by $- 1$ .

$\sqrt{(1 + 4 x) (1 - 4 x)}$