Trigonometry Examples

$\sqrt{\frac{\sin^{2} (y)}{2 \cos^{2} (y)}}$

Step 1

Multiply by $1$ .

$\sqrt{\frac{\sin^{2} (y) \cdot 1}{2 \cos^{2} (y)}}$

Step 2

Multiply by $1$ .

$\sqrt{\frac{\sin^{2} (y) \cdot 1}{2 (\cos^{2} (y) \cdot 1)}}$

Step 3

Separate fractions.

$\sqrt{\frac{1}{2 \cdot (1)} \cdot \frac{\sin^{2} (y)}{\cos^{2} (y)}}$

Step 4

Convert from $\frac{\sin^{2} (y)}{\cos^{2} (y)}$ to $\tan^{2} (y)$ .

$\sqrt{\frac{1}{2 \cdot (1)} \tan^{2} (y)}$

Step 5

Multiply $2$ by $1$ .

$\sqrt{\frac{1}{2} \tan^{2} (y)}$

Step 6

Combine $\frac{1}{2}$ and $\tan^{2} (y)$ .

$\sqrt{\frac{\tan^{2} (y)}{2}}$

Step 7

Rewrite $\sqrt{\frac{\tan^{2} (y)}{2}}$ as $\frac{\sqrt{\tan^{2} (y)}}{\sqrt{2}}$ .

$\frac{\sqrt{\tan^{2} (y)}}{\sqrt{2}}$

Step 8

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

$\frac{\tan (y)}{\sqrt{2}}$

Step 9

Multiply $\frac{\tan (y)}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\tan (y)}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{\tan (y)}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\tan (y) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 10.2

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

$\frac{\tan (y) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 10.3

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

$\frac{\tan (y) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 10.4

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

$\frac{\tan (y) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 10.5

Add $1$ and $1$ .

$\frac{\tan (y) \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 10.6

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

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Step 10.6.1

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

$\frac{\tan (y) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 10.6.2

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

$\frac{\tan (y) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 10.6.3

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

$\frac{\tan (y) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$\frac{\tan (y) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 10.6.4.2

Rewrite the expression.

$\frac{\tan (y) \sqrt{2}}{2^{1}}$

Step 10.6.5

Evaluate the exponent.

$\frac{\tan (y) \sqrt{2}}{2}$