Trigonometry Examples

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

Step 1

Multiply $- (- \frac{2}{3})$ .

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Step 1.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2

Multiply $\frac{2}{3}$ by $1$ .

$\sqrt{\frac{1 + \frac{2}{3}}{2}}$

Step 2

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Add $3$ and $2$ .

$\sqrt{\frac{\frac{5}{3}}{2}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{5}{3} \cdot \frac{1}{2}}$

Step 6

Multiply $\frac{5}{3} \cdot \frac{1}{2}$ .

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Step 6.1

Multiply $\frac{5}{3}$ by $\frac{1}{2}$ .

$\sqrt{\frac{5}{3 \cdot 2}}$

Step 6.2

Multiply $3$ by $2$ .

$\sqrt{\frac{5}{6}}$

Step 7

Rewrite $\sqrt{\frac{5}{6}}$ as $\frac{\sqrt{5}}{\sqrt{6}}$ .

$\frac{\sqrt{5}}{\sqrt{6}}$

Step 8

Multiply $\frac{\sqrt{5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{5}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt{5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{5} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 9.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 9.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 9.4

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

$\frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$\frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 9.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{5} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 9.6.2

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

$\frac{\sqrt{5} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

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

$\frac{\sqrt{5} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

$\frac{\sqrt{5} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$\frac{\sqrt{5} \sqrt{6}}{6^{1}}$

Step 9.6.5

Evaluate the exponent.

$\frac{\sqrt{5} \sqrt{6}}{6}$

Step 10

Simplify the numerator.

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Step 10.1

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 6}}{6}$

Step 10.2

Multiply $5$ by $6$ .

$\frac{\sqrt{30}}{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{30}}{6}$

Decimal Form:

$0.91287092 \dots$