Trigonometry Examples

$\sqrt{1 - {(- \frac{9}{41})}^{2}}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $- \frac{9}{41}$ .

$\sqrt{1 - ({(- 1)}^{2} {(\frac{9}{41})}^{2})}$

Step 1.2

Apply the product rule to $\frac{9}{41}$ .

$\sqrt{1 - ({(- 1)}^{2} \frac{9^{2}}{41^{2}})}$

Step 2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.1

Move ${(- 1)}^{2}$ .

$\sqrt{1 + {(- 1)}^{2} \cdot - 1 \frac{9^{2}}{41^{2}}}$

Step 2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 2.2.1

Raise $- 1$ to the power of $1$ .

$\sqrt{1 + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{9^{2}}{41^{2}}}$

Step 2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{1 + {(- 1)}^{2 + 1} \frac{9^{2}}{41^{2}}}$

Step 2.3

Add $2$ and $1$ .

$\sqrt{1 + {(- 1)}^{3} \frac{9^{2}}{41^{2}}}$

Step 3

Raise $- 1$ to the power of $3$ .

$\sqrt{1 - \frac{9^{2}}{41^{2}}}$

Step 4

Raise $9$ to the power of $2$ .

$\sqrt{1 - \frac{81}{41^{2}}}$

Step 5

Raise $41$ to the power of $2$ .

$\sqrt{1 - \frac{81}{1681}}$

Step 6

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

$\sqrt{\frac{1681}{1681} - \frac{81}{1681}}$

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{1681 - 81}{1681}}$

Step 8

Subtract $81$ from $1681$ .

$\sqrt{\frac{1600}{1681}}$

Step 9

Rewrite $\sqrt{\frac{1600}{1681}}$ as $\frac{\sqrt{1600}}{\sqrt{1681}}$ .

$\frac{\sqrt{1600}}{\sqrt{1681}}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite $1600$ as $40^{2}$ .

$\frac{\sqrt{40^{2}}}{\sqrt{1681}}$

Step 10.2

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

$\frac{40}{\sqrt{1681}}$

Step 11

Simplify the denominator.

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Step 11.1

Rewrite $1681$ as $41^{2}$ .

$\frac{40}{\sqrt{41^{2}}}$

Step 11.2

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

$\frac{40}{41}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{40}{41}$

Decimal Form:

$0.\overline{97560}$