Trigonometry Examples

$\sqrt{1 - \frac{\frac{1}{25}}{2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{1 - (\frac{1}{25} \cdot \frac{1}{2})}$

Step 2

Multiply $\frac{1}{25} \cdot \frac{1}{2}$ .

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Step 2.1

Multiply $\frac{1}{25}$ by $\frac{1}{2}$ .

$\sqrt{1 - \frac{1}{25 \cdot 2}}$

Step 2.2

Multiply $25$ by $2$ .

$\sqrt{1 - \frac{1}{50}}$

Step 3

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

$\sqrt{\frac{50}{50} - \frac{1}{50}}$

Step 4

Combine the numerators over the common denominator.

$\sqrt{\frac{50 - 1}{50}}$

Step 5

Subtract $1$ from $50$ .

$\sqrt{\frac{49}{50}}$

Step 6

Rewrite $\sqrt{\frac{49}{50}}$ as $\frac{\sqrt{49}}{\sqrt{50}}$ .

$\frac{\sqrt{49}}{\sqrt{50}}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{7^{2}}}{\sqrt{50}}$

Step 7.2

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

$\frac{7}{\sqrt{50}}$

Step 8

Simplify the denominator.

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Step 8.1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 8.1.1

Factor $25$ out of $50$ .

$\frac{7}{\sqrt{25 (2)}}$

Step 8.1.2

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

$\frac{7}{\sqrt{5^{2} \cdot 2}}$

Step 8.2

Pull terms out from under the radical.

$\frac{7}{5 \sqrt{2}}$

Step 9

Multiply $\frac{7}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{7}{5 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{7}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{7 \sqrt{2}}{5 \sqrt{2} \sqrt{2}}$

Step 10.2

Move $\sqrt{2}$ .

$\frac{7 \sqrt{2}}{5 (\sqrt{2} \sqrt{2})}$

Step 10.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{7 \sqrt{2}}{5 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 10.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{7 \sqrt{2}}{5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 10.5

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

$\frac{7 \sqrt{2}}{5 {\sqrt{2}}^{1 + 1}}$

Step 10.6

Add $1$ and $1$ .

$\frac{7 \sqrt{2}}{5 {\sqrt{2}}^{2}}$

Step 10.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 10.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{7 \sqrt{2}}{5 {(2^{\frac{1}{2}})}^{2}}$

Step 10.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{7 \sqrt{2}}{5 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 10.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{7 \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 10.7.4

Cancel the common factor of $2$ .

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Step 10.7.4.1

Cancel the common factor.

$\frac{7 \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 10.7.4.2

Rewrite the expression.

$\frac{7 \sqrt{2}}{5 \cdot 2^{1}}$

Step 10.7.5

Evaluate the exponent.

$\frac{7 \sqrt{2}}{5 \cdot 2}$

Step 11

Multiply $5$ by $2$ .

$\frac{7 \sqrt{2}}{10}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{7 \sqrt{2}}{10}$

Decimal Form:

$0.98994949 \dots$