Trigonometry Examples

$\sqrt{1 - {(\frac{12}{13})}^{2}}$

Step 1

Apply the product rule to $\frac{12}{13}$ .

$\sqrt{1 - \frac{12^{2}}{13^{2}}}$

Step 2

Raise $12$ to the power of $2$ .

$\sqrt{1 - \frac{144}{13^{2}}}$

Step 3

Raise $13$ to the power of $2$ .

$\sqrt{1 - \frac{144}{169}}$

Step 4

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

$\sqrt{\frac{169}{169} - \frac{144}{169}}$

Step 5

Combine the numerators over the common denominator.

$\sqrt{\frac{169 - 144}{169}}$

Step 6

Subtract $144$ from $169$ .

$\sqrt{\frac{25}{169}}$

Step 7

Rewrite $\sqrt{\frac{25}{169}}$ as $\frac{\sqrt{25}}{\sqrt{169}}$ .

$\frac{\sqrt{25}}{\sqrt{169}}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{\sqrt{169}}$

Step 8.2

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

$\frac{5}{\sqrt{169}}$

Step 9

Simplify the denominator.

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Step 9.1

Rewrite $169$ as $13^{2}$ .

$\frac{5}{\sqrt{13^{2}}}$

Step 9.2

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

$\frac{5}{13}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{13}$

Decimal Form:

$0.\overline{384615}$