Trigonometry Examples

$\sqrt{{(\frac{3}{8})}^{2} - 1}$

Step 1

Apply the product rule to $\frac{3}{8}$ .

$\sqrt{\frac{3^{2}}{8^{2}} - 1}$

Step 2

Raise $3$ to the power of $2$ .

$\sqrt{\frac{9}{8^{2}} - 1}$

Step 3

Raise $8$ to the power of $2$ .

$\sqrt{\frac{9}{64} - 1}$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$\sqrt{\frac{9}{64} - 1 \cdot \frac{64}{64}}$

Step 5

Combine $- 1$ and $\frac{64}{64}$ .

$\sqrt{\frac{9}{64} + \frac{- 1 \cdot 64}{64}}$

Step 6

Combine the numerators over the common denominator.

$\sqrt{\frac{9 - 1 \cdot 64}{64}}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $64$ .

$\sqrt{\frac{9 - 64}{64}}$

Step 7.2

Subtract $64$ from $9$ .

$\sqrt{\frac{- 55}{64}}$

Step 8

Move the negative in front of the fraction.

$\sqrt{- \frac{55}{64}}$

Step 9

Rewrite $- \frac{55}{64}$ as ${(\frac{i}{8})}^{2} \cdot 55$ .

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Step 9.1

Rewrite $- 1$ as $i^{2}$ .

$\sqrt{i^{2} \frac{55}{64}}$

Step 9.2

Factor the perfect power $1^{2}$ out of $55$ .

$\sqrt{i^{2} \frac{1^{2} \cdot 55}{64}}$

Step 9.3

Factor the perfect power $8^{2}$ out of $64$ .

$\sqrt{i^{2} \frac{1^{2} \cdot 55}{8^{2} \cdot 1}}$

Step 9.4

Rearrange the fraction $\frac{1^{2} \cdot 55}{8^{2} \cdot 1}$ .

$\sqrt{i^{2} ({(\frac{1}{8})}^{2} \cdot 55)}$

Step 9.5

Rewrite $i^{2} {(\frac{1}{8})}^{2}$ as ${(\frac{i}{8})}^{2}$ .

$\sqrt{{(\frac{i}{8})}^{2} \cdot 55}$

Step 10

Pull terms out from under the radical.

$\frac{i}{8} \sqrt{55}$

Step 11

Combine $\frac{i}{8}$ and $\sqrt{55}$ .

$\frac{i \sqrt{55}}{8}$