Calculus Examples

$\sin (2 \arccos (x))$

Step 1

Apply the sine double-angle identity.

$2 \sin (\arccos (x)) \cos (\arccos (x))$

Step 2

Write the expression using exponents.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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 $b = x$ .

$2 \sqrt{(1 + x) (1 - x)} \cos (\arccos (x))$

Step 4

The functions cosine and arccosine are inverses.

$2 \sqrt{(1 + x) (1 - x)} x$