Calculus Examples

$\cos (\arcsin (\frac{x}{3}))$

Step 1

Write the expression using exponents.

Tap for more steps...

Step 1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{x}{3})}^{2}}, \frac{x}{3})$ , $(\sqrt{1^{2} - {(\frac{x}{3})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{x}{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{x}{3})}^{2}}, \frac{x}{3})$ . Therefore, $\cos (\arcsin (\frac{x}{3}))$ is $\sqrt{1 - {(\frac{x}{3})}^{2}}$ .

$\sqrt{1 - {(\frac{x}{3})}^{2}}$

Step 1.2

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

$\sqrt{1^{2} - {(\frac{x}{3})}^{2}}$

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 $b = \frac{x}{3}$ .

$\sqrt{(1 + \frac{x}{3}) (1 - \frac{x}{3})}$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Write $1$ as a fraction with a common denominator.

$\sqrt{(\frac{3}{3} + \frac{x}{3}) (1 - \frac{x}{3})}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{\frac{3 + x}{3} (1 - \frac{x}{3})}$

Step 3.3

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{3 + x}{3} (\frac{3}{3} - \frac{x}{3})}$

Step 3.4

Combine the numerators over the common denominator.

$\sqrt{\frac{3 + x}{3} \cdot \frac{3 - x}{3}}$

Step 3.5

Multiply $\frac{3 + x}{3}$ by $\frac{3 - x}{3}$ .

$\sqrt{\frac{(3 + x) (3 - x)}{3 \cdot 3}}$

Step 3.6

Multiply $3$ by $3$ .

$\sqrt{\frac{(3 + x) (3 - x)}{9}}$

Step 4

Rewrite $\frac{(3 + x) (3 - x)}{9}$ as ${(\frac{1}{3})}^{2} ((3 + x) (3 - x))$ .

Tap for more steps...

Step 4.1

Factor the perfect power $1^{2}$ out of $(3 + x) (3 - x)$ .

$\sqrt{\frac{1^{2} ((3 + x) (3 - x))}{9}}$

Step 4.2

Factor the perfect power $3^{2}$ out of $9$ .

$\sqrt{\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}}$

Step 4.3

Rearrange the fraction $\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{3})}^{2} ((3 + x) (3 - x))}$

Step 5

Pull terms out from under the radical.

$\frac{1}{3} \sqrt{(3 + x) (3 - x)}$

Step 6

Combine $\frac{1}{3}$ and $\sqrt{(3 + x) (3 - x)}$ .

$\frac{\sqrt{(3 + x) (3 - x)}}{3}$