Trigonometry Examples

\sqrt{\frac{1 - (\frac{10}{24})}{2}}

$\sqrt{\frac{1 - (\frac{10}{24})}{2}}$

Step 1

Cancel the common factor of

10

$10$ and

24

$24$ .

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Step 1.1

Factor

2

$2$ out of

10

$10$ .

\sqrt{\frac{1 - \frac{2 (5)}{24}}{2}}

$\sqrt{\frac{1 - \frac{2 (5)}{24}}{2}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor

2

$2$ out of

24

$24$ .

\sqrt{\frac{1 - \frac{2 \cdot 5}{2 \cdot 12}}{2}}

$\sqrt{\frac{1 - \frac{2 \cdot 5}{2 \cdot 12}}{2}}$

Step 1.2.2

Cancel the common factor.

$\sqrt{\frac{1 - \frac{2 \cdot 5}{2 \cdot 12}}{2}}$

Step 1.2.3

Rewrite the expression.

$\sqrt{\frac{1 - \frac{5}{12}}{2}}$

Step 2

Simplify the expression.

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Step 2.1

Write

$1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{12}{12} - \frac{5}{12}}{2}}$

Step 2.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{12 - 5}{12}}{2}}$

Step 2.3

Subtract

$5$ from

$12$ .

$\sqrt{\frac{\frac{7}{12}}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{7}{12} \cdot \frac{1}{2}}$

Step 4

Multiply

$\frac{7}{12} \cdot \frac{1}{2}$ .

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Step 4.1

Multiply

$\frac{7}{12}$ by

$\frac{1}{2}$ .

$\sqrt{\frac{7}{12 \cdot 2}}$

Step 4.2

Multiply

$12$ by

$2$ .

$\sqrt{\frac{7}{24}}$

Step 5

Rewrite

$\sqrt{\frac{7}{24}}$ as

$\frac{\sqrt{7}}{\sqrt{24}}$ .

$\frac{\sqrt{7}}{\sqrt{24}}$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite

$24$ as

$2^{2} \cdot 6$ .

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Step 6.1.1

Factor

$4$ out of

$24$ .

$\frac{\sqrt{7}}{\sqrt{4 (6)}}$

Step 6.1.2

Rewrite

$4$ as

$2^{2}$ .

$\frac{\sqrt{7}}{\sqrt{2^{2} \cdot 6}}$

Step 6.2

Pull terms out from under the radical.

$\frac{\sqrt{7}}{2 \sqrt{6}}$

Step 7

Multiply

$\frac{\sqrt{7}}{2 \sqrt{6}}$ by

$\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{7}}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply

$\frac{\sqrt{7}}{2 \sqrt{6}}$ by

$\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{7} \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Step 8.2

Move

$\sqrt{6}$ .

$\frac{\sqrt{7} \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 8.3

Raise

$\sqrt{6}$ to the power of

$1$ .

$\frac{\sqrt{7} \sqrt{6}}{2 ({\sqrt{6}}^{1} \sqrt{6})}$

Step 8.4

Raise

$\sqrt{6}$ to the power of

$1$ .

$\frac{\sqrt{7} \sqrt{6}}{2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

Step 8.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{7} \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Step 8.6

Add

$1$ and

$1$ .

$\frac{\sqrt{7} \sqrt{6}}{2 {\sqrt{6}}^{2}}$

Step 8.7

Rewrite

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

$6$ .

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Step 8.7.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{6}$ as

$6^{\frac{1}{2}}$ .

$\frac{\sqrt{7} \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

Step 8.7.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{7} \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 8.7.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{\sqrt{7} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 8.7.4

Cancel the common factor of

$2$ .

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Step 8.7.4.1

Cancel the common factor.

$\frac{\sqrt{7} \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 8.7.4.2

Rewrite the expression.

$\frac{\sqrt{7} \sqrt{6}}{2 \cdot 6^{1}}$

Step 8.7.5

Evaluate the exponent.

$\frac{\sqrt{7} \sqrt{6}}{2 \cdot 6}$

Step 9

Simplify the numerator.

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Step 9.1

Combine using the product rule for radicals.

$\frac{\sqrt{7 \cdot 6}}{2 \cdot 6}$

Step 9.2

Multiply

$7$ by

$6$ .

$\frac{\sqrt{42}}{2 \cdot 6}$

Step 10

Multiply

$2$ by

$6$ .

$\frac{\sqrt{42}}{12}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{42}}{12}$

Decimal Form:

$0.54006172 \dots$