Trigonometry Examples

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

Step 1

Apply the product rule to $\frac{1}{3}$ .

$\sqrt{1 - \frac{1^{2}}{3^{2}}}$

Step 2

One to any power is one.

$\sqrt{1 - \frac{1}{3^{2}}}$

Step 3

Raise $3$ to the power of $2$ .

$\sqrt{1 - \frac{1}{9}}$

Step 4

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

$\sqrt{\frac{9}{9} - \frac{1}{9}}$

Step 5

Combine the numerators over the common denominator.

$\sqrt{\frac{9 - 1}{9}}$

Step 6

Subtract $1$ from $9$ .

$\sqrt{\frac{8}{9}}$

Step 7

Rewrite $\sqrt{\frac{8}{9}}$ as $\frac{\sqrt{8}}{\sqrt{9}}$ .

$\frac{\sqrt{8}}{\sqrt{9}}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 8.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)}}{\sqrt{9}}$

Step 8.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 2}}{\sqrt{9}}$

Step 8.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2}}{\sqrt{9}}$

Step 9

Simplify the denominator.

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Step 9.1

Rewrite $9$ as $3^{2}$ .

$\frac{2 \sqrt{2}}{\sqrt{3^{2}}}$

Step 9.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2 \sqrt{2}}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{2}}{3}$

Decimal Form:

$0.94280904 \dots$