Trigonometry Examples

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

Step 1

Cancel the common factor of $10$ and $24$ .

Tap for more steps...

Step 1.1

Factor $2$ out of $10$ .

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

Step 1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.1

Factor $2$ out of $24$ .

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

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.

Tap for more steps...

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

Tap for more steps...

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.

Tap for more steps...

Step 6.1

Rewrite $24$ as $2^{2} \cdot 6$ .

Tap for more steps...

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.

Tap for more steps...

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 $6$ .

Tap for more steps...

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 $2$ .

Tap for more steps...

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.

Tap for more steps...

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:

$0.54006172 \dots$