Precalculus Examples

$- \sqrt{\frac{1 - \cos (225)}{1 + \cos (225)}}$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$- \sqrt{\frac{1 - - \cos (45)}{1 + \cos (225)}}$

Step 2

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$- \sqrt{\frac{1 - - \frac{\sqrt{2}}{2}}{1 + \cos (225)}}$

Step 3

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 3.1

Multiply $- 1$ by $- 1$ .

$- \sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{1 + \cos (225)}}$

Step 3.2

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

$- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{1 + \cos (225)}}$

Step 4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{1 - \cos (45)}}$

Step 5

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

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

Step 6

Multiply the numerator and denominator of the fraction by $2$ .

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Step 6.1

Multiply $\frac{1 + \frac{\sqrt{2}}{2}}{1 - \frac{\sqrt{2}}{2}}$ by $\frac{2}{2}$ .

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

Step 6.2

Combine.

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

Step 7

Apply the distributive property.

$- \sqrt{\frac{2 \cdot 1 + 2 \frac{\sqrt{2}}{2}}{2 \cdot 1 + 2 (- \frac{\sqrt{2}}{2})}}$

Step 8

Simplify by cancelling.

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

Cancel the common factor.

$- \sqrt{\frac{2 \cdot 1 + 2 \frac{\sqrt{2}}{2}}{2 \cdot 1 + 2 (- \frac{\sqrt{2}}{2})}}$

Step 8.1.2

Rewrite the expression.

$- \sqrt{\frac{2 \cdot 1 + \sqrt{2}}{2 \cdot 1 + 2 (- \frac{\sqrt{2}}{2})}}$

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

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

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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

Step 9

Simplify.

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Step 9.1

Multiply $2$ by $1$ .

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

Step 9.2

Multiply $2$ by $1$ .

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

Step 10

Multiply $\frac{2 + \sqrt{2}}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

$- \sqrt{\frac{2 + \sqrt{2}}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}}}$

Step 11

Combine fractions.

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Step 11.1

Multiply $\frac{2 + \sqrt{2}}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

$- \sqrt{\frac{(2 + \sqrt{2}) (2 + \sqrt{2})}{(2 - \sqrt{2}) (2 + \sqrt{2})}}$

Step 11.2

Expand the denominator using the FOIL method.

$- \sqrt{\frac{(2 + \sqrt{2}) (2 + \sqrt{2})}{4 + 2 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{2}}}$

Step 11.3

Simplify.

$- \sqrt{\frac{(2 + \sqrt{2}) (2 + \sqrt{2})}{2}}$

Step 12

Simplify the numerator.

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Add $1$ and $1$ .

$- \sqrt{\frac{{(2 + \sqrt{2})}^{2}}{2}}$

Step 13

Rewrite ${(2 + \sqrt{2})}^{2}$ as $(2 + \sqrt{2}) (2 + \sqrt{2})$ .

$- \sqrt{\frac{(2 + \sqrt{2}) (2 + \sqrt{2})}{2}}$

Step 14

Expand $(2 + \sqrt{2}) (2 + \sqrt{2})$ using the FOIL Method.

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Step 14.1

Apply the distributive property.

$- \sqrt{\frac{2 (2 + \sqrt{2}) + \sqrt{2} (2 + \sqrt{2})}{2}}$

Step 14.2

Apply the distributive property.

$- \sqrt{\frac{2 \cdot 2 + 2 \sqrt{2} + \sqrt{2} (2 + \sqrt{2})}{2}}$

Step 14.3

Apply the distributive property.

$- \sqrt{\frac{2 \cdot 2 + 2 \sqrt{2} + \sqrt{2} \cdot 2 + \sqrt{2} \sqrt{2}}{2}}$

Step 15

Simplify and combine like terms.

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Step 15.1

Simplify each term.

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Step 15.1.1

Multiply $2$ by $2$ .

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

Step 15.1.2

Move $2$ to the left of $\sqrt{2}$ .

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

Step 15.1.3

Combine using the product rule for radicals.

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

Step 15.1.4

Multiply $2$ by $2$ .

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

Step 15.1.5

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

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

Step 15.1.6

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

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

Step 15.2

Add $4$ and $2$ .

$- \sqrt{\frac{6 + 2 \sqrt{2} + 2 \sqrt{2}}{2}}$

Step 15.3

Add $2 \sqrt{2}$ and $2 \sqrt{2}$ .

$- \sqrt{\frac{6 + 4 \sqrt{2}}{2}}$

Step 16

Cancel the common factor of $6 + 4 \sqrt{2}$ and $2$ .

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Step 16.1

Factor $2$ out of $6$ .

$- \sqrt{\frac{2 \cdot 3 + 4 \sqrt{2}}{2}}$

Step 16.2

Factor $2$ out of $4 \sqrt{2}$ .

$- \sqrt{\frac{2 \cdot 3 + 2 (2 \sqrt{2})}{2}}$

Step 16.3

Factor $2$ out of $2 (3) + 2 (2 \sqrt{2})$ .

$- \sqrt{\frac{2 (3 + 2 \sqrt{2})}{2}}$

Step 16.4

Cancel the common factors.

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Step 16.4.1

Factor $2$ out of $2$ .

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

Step 16.4.2

Cancel the common factor.

$- \sqrt{\frac{2 (3 + 2 \sqrt{2})}{2 \cdot 1}}$

Step 16.4.3

Rewrite the expression.

$- \sqrt{\frac{3 + 2 \sqrt{2}}{1}}$

Step 16.4.4

Divide $3 + 2 \sqrt{2}$ by $1$ .

$- \sqrt{3 + 2 \sqrt{2}}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{3 + 2 \sqrt{2}}$

Decimal Form:

$- 2.41421356 \dots$