Trigonometry Examples

$\frac{- \frac{1}{2}}{\sqrt{1 \frac{1}{4}}}$

Step 1

Convert $1 \frac{1}{4}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$\frac{- \frac{1}{2}}{\sqrt{1 + \frac{1}{4}}}$

Step 1.2

Add $1$ and $\frac{1}{4}$ .

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Step 1.2.1

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

$\frac{- \frac{1}{2}}{\sqrt{\frac{4}{4} + \frac{1}{4}}}$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{- \frac{1}{2}}{\sqrt{\frac{4 + 1}{4}}}$

Step 1.2.3

Add $4$ and $1$ .

$\frac{- \frac{1}{2}}{\sqrt{\frac{5}{4}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{2} \cdot \frac{1}{\sqrt{\frac{5}{4}}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $\sqrt{\frac{5}{4}}$ as $\frac{\sqrt{5}}{\sqrt{4}}$ .

$- \frac{1}{2} \cdot \frac{1}{\frac{\sqrt{5}}{\sqrt{4}}}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.2

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$- (\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 8.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 8.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Evaluate the exponent.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.44721359 \dots$