Trigonometry Examples

$\frac{\tan (112.5)}{1 - \tan^{2} (112.5)}$

Step 1

Evaluate $\tan (112.5)$ .

$\frac{- 2.41421356}{1 - \tan^{2} (112.5)}$

Step 2

Simplify the denominator.

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Step 2.1

The exact value of $\tan (112.5)$ is $- \sqrt{3 + 2 \sqrt{2}}$ .

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Step 2.1.1

Rewrite $112.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\frac{- 2.41421356}{1 - \tan^{2} (\frac{225}{2})}$

Step 2.1.2

Apply the tangent half-angle identity.

$\frac{- 2.41421356}{1 - {(\pm \sqrt{\frac{1 - \cos (225)}{1 + \cos (225)}})}^{2}}$

Step 2.1.3

Change the $\pm$ to $-$ because tangent is negative in the second quadrant.

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{1 - \cos (225)}{1 + \cos (225)}})}^{2}}$

Step 2.1.4

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

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Step 2.1.4.1

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.

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{1 - - \cos (45)}{1 + \cos (225)}})}^{2}}$

Step 2.1.4.2

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

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{1 - - \frac{\sqrt{2}}{2}}{1 + \cos (225)}})}^{2}}$

Step 2.1.4.3

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

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Step 2.1.4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{1 + \cos (225)}})}^{2}}$

Step 2.1.4.3.2

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

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{1 + \cos (225)}})}^{2}}$

Step 2.1.4.4

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

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{1 + \cos (225)}})}^{2}}$

Step 2.1.4.5

Combine the numerators over the common denominator.

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 + \cos (225)}})}^{2}}$

Step 2.1.4.6

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.

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 - \cos (45)}})}^{2}}$

Step 2.1.4.7

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

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 - \frac{\sqrt{2}}{2}}})}^{2}}$

Step 2.1.4.8

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

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{\frac{2}{2} - \frac{\sqrt{2}}{2}}})}^{2}}$

Step 2.1.4.9

Combine the numerators over the common denominator.

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{\frac{2 - \sqrt{2}}{2}}})}^{2}}$

Step 2.1.4.10

Multiply the numerator by the reciprocal of the denominator.

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{2}{2 - \sqrt{2}}})}^{2}}$

Step 2.1.4.11

Cancel the common factor of $2$ .

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Step 2.1.4.11.1

Cancel the common factor.

$\frac{- 2.41421356}{1 - {(- \sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{2}{2 - \sqrt{2}}})}^{2}}$

Step 2.1.4.11.2

Rewrite the expression.

$\frac{- 2.41421356}{1 - {(- \sqrt{(2 + \sqrt{2}) \frac{1}{2 - \sqrt{2}}})}^{2}}$

Step 2.1.4.12

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

$\frac{- 2.41421356}{1 - {(- \sqrt{(2 + \sqrt{2}) (\frac{1}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}})})}^{2}}$

Step 2.1.4.13

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

$\frac{- 2.41421356}{1 - {(- \sqrt{(2 + \sqrt{2}) \frac{2 + \sqrt{2}}{(2 - \sqrt{2}) (2 + \sqrt{2})}})}^{2}}$

Step 2.1.4.14

Expand the denominator using the FOIL method.

$\frac{- 2.41421356}{1 - {(- \sqrt{(2 + \sqrt{2}) \frac{2 + \sqrt{2}}{4 + 2 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{2}}})}^{2}}$

Step 2.1.4.15

Simplify.

$\frac{- 2.41421356}{1 - {(- \sqrt{(2 + \sqrt{2}) \frac{2 + \sqrt{2}}{2}})}^{2}}$

Step 2.1.4.16

Apply the distributive property.

$\frac{- 2.41421356}{1 - {(- \sqrt{2 \frac{2 + \sqrt{2}}{2} + \sqrt{2} \frac{2 + \sqrt{2}}{2}})}^{2}}$

Step 2.1.4.17

Cancel the common factor of $2$ .

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Step 2.1.4.17.1

Cancel the common factor.

$\frac{- 2.41421356}{1 - {(- \sqrt{2 \frac{2 + \sqrt{2}}{2} + \sqrt{2} \frac{2 + \sqrt{2}}{2}})}^{2}}$

Step 2.1.4.17.2

Rewrite the expression.

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \sqrt{2} \frac{2 + \sqrt{2}}{2}})}^{2}}$

Step 2.1.4.18

Combine $\sqrt{2}$ and $\frac{2 + \sqrt{2}}{2}$ .

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{\sqrt{2} (2 + \sqrt{2})}{2}})}^{2}}$

Step 2.1.4.19

Simplify each term.

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Step 2.1.4.19.1

Apply the distributive property.

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{\sqrt{2} \cdot 2 + \sqrt{2} \sqrt{2}}{2}})}^{2}}$

Step 2.1.4.19.2

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

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 \cdot \sqrt{2} + \sqrt{2} \sqrt{2}}{2}})}^{2}}$

Step 2.1.4.19.3

Combine using the product rule for radicals.

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 \cdot \sqrt{2} + \sqrt{2 \cdot 2}}{2}})}^{2}}$

Step 2.1.4.19.4

Simplify each term.

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Step 2.1.4.19.4.1

Multiply $2$ by $2$ .

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 \sqrt{2} + \sqrt{4}}{2}})}^{2}}$

Step 2.1.4.19.4.2

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

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 \sqrt{2} + \sqrt{2^{2}}}{2}})}^{2}}$

Step 2.1.4.19.4.3

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

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 \sqrt{2} + 2}{2}})}^{2}}$

Step 2.1.4.19.5

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

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Step 2.1.4.19.5.1

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

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 (\sqrt{2}) + 2}{2}})}^{2}}$

Step 2.1.4.19.5.2

Factor $2$ out of $2$ .

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 (\sqrt{2}) + 2 \cdot 1}{2}})}^{2}}$

Step 2.1.4.19.5.3

Factor $2$ out of $2 (\sqrt{2}) + 2 (1)$ .

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 (\sqrt{2} + 1)}{2}})}^{2}}$

Step 2.1.4.19.5.4

Cancel the common factors.

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Step 2.1.4.19.5.4.1

Factor $2$ out of $2$ .

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 (\sqrt{2} + 1)}{2 (1)}})}^{2}}$

Step 2.1.4.19.5.4.2

Cancel the common factor.

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{2 (\sqrt{2} + 1)}{2 \cdot 1}})}^{2}}$

Step 2.1.4.19.5.4.3

Rewrite the expression.

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \frac{\sqrt{2} + 1}{1}})}^{2}}$

Step 2.1.4.19.5.4.4

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

$\frac{- 2.41421356}{1 - {(- \sqrt{2 + \sqrt{2} + \sqrt{2} + 1})}^{2}}$

Step 2.1.4.20

Add $2$ and $1$ .

$\frac{- 2.41421356}{1 - {(- \sqrt{3 + \sqrt{2} + \sqrt{2}})}^{2}}$

Step 2.1.4.21

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

$\frac{- 2.41421356}{1 - {(- \sqrt{3 + 2 \sqrt{2}})}^{2}}$

Step 2.2

Apply the product rule to $- \sqrt{3 + 2 \sqrt{2}}$ .

$\frac{- 2.41421356}{1 - ({(- 1)}^{2} {\sqrt{3 + 2 \sqrt{2}}}^{2})}$

Step 2.3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.3.1

Move ${(- 1)}^{2}$ .

$\frac{- 2.41421356}{1 + {(- 1)}^{2} \cdot - 1 {\sqrt{3 + 2 \sqrt{2}}}^{2}}$

Step 2.3.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 2.3.2.1

Raise $- 1$ to the power of $1$ .

$\frac{- 2.41421356}{1 + {(- 1)}^{2} \cdot {(- 1)}^{1} {\sqrt{3 + 2 \sqrt{2}}}^{2}}$

Step 2.3.2.2

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

$\frac{- 2.41421356}{1 + {(- 1)}^{2 + 1} {\sqrt{3 + 2 \sqrt{2}}}^{2}}$

Step 2.3.3

Add $2$ and $1$ .

$\frac{- 2.41421356}{1 + {(- 1)}^{3} {\sqrt{3 + 2 \sqrt{2}}}^{2}}$

Step 2.4

Raise $- 1$ to the power of $3$ .

$\frac{- 2.41421356}{1 - {\sqrt{3 + 2 \sqrt{2}}}^{2}}$

Step 2.5

Rewrite ${\sqrt{3 + 2 \sqrt{2}}}^{2}$ as $3 + 2 \sqrt{2}$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 + 2 \sqrt{2}}$ as ${(3 + 2 \sqrt{2})}^{\frac{1}{2}}$ .

$\frac{- 2.41421356}{1 - {({(3 + 2 \sqrt{2})}^{\frac{1}{2}})}^{2}}$

Step 2.5.2

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

$\frac{- 2.41421356}{1 - {(3 + 2 \sqrt{2})}^{\frac{1}{2} \cdot 2}}$

Step 2.5.3

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

$\frac{- 2.41421356}{1 - {(3 + 2 \sqrt{2})}^{\frac{2}{2}}}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{- 2.41421356}{1 - {(3 + 2 \sqrt{2})}^{\frac{2}{2}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{- 2.41421356}{1 - {(3 + 2 \sqrt{2})}^{1}}$

Step 2.5.5

Simplify.

$\frac{- 2.41421356}{1 - (3 + 2 \sqrt{2})}$

Step 2.6

Apply the distributive property.

$\frac{- 2.41421356}{1 - 1 \cdot 3 - (2 \sqrt{2})}$

Step 2.7

Multiply $- 1$ by $3$ .

$\frac{- 2.41421356}{1 - 3 - (2 \sqrt{2})}$

Step 2.8

Multiply $2$ by $- 1$ .

$\frac{- 2.41421356}{1 - 3 - 2 \sqrt{2}}$

Step 2.9

Subtract $3$ from $1$ .

$\frac{- 2.41421356}{- 2 - 2 \sqrt{2}}$

Step 3

Move the negative in front of the fraction.

$- \frac{2.41421356}{- 2 - 2 \sqrt{2}}$

Step 4

Multiply $\frac{2.41421356}{- 2 - 2 \sqrt{2}}$ by $\frac{- 2 + 2 \sqrt{2}}{- 2 + 2 \sqrt{2}}$ .

$- (\frac{2.41421356}{- 2 - 2 \sqrt{2}} \cdot \frac{- 2 + 2 \sqrt{2}}{- 2 + 2 \sqrt{2}})$

Step 5

Multiply $\frac{2.41421356}{- 2 - 2 \sqrt{2}}$ by $\frac{- 2 + 2 \sqrt{2}}{- 2 + 2 \sqrt{2}}$ .

$- \frac{2.41421356 (- 2 + 2 \sqrt{2})}{(- 2 - 2 \sqrt{2}) (- 2 + 2 \sqrt{2})}$

Step 6

Expand the denominator using the FOIL method.

$- \frac{2.41421356 (- 2 + 2 \sqrt{2})}{4 - 4 \sqrt{2} + 4 \sqrt{2} - 4 {\sqrt{2}}^{2}}$

Step 7

Simplify.

$- \frac{2.41421356 (- 2 + 2 \sqrt{2})}{- 4}$

Step 8

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

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Step 8.1

Factor $2$ out of $2.41421356 (- 2 + 2 \sqrt{2})$ .

$- \frac{2 (1.20710678 (- 2 + 2 \sqrt{2}))}{- 4}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $- 4$ .

$- \frac{2 (1.20710678 (- 2 + 2 \sqrt{2}))}{2 \cdot - 2}$

Step 8.2.2

Cancel the common factor.

$- \frac{2 (1.20710678 (- 2 + 2 \sqrt{2}))}{2 \cdot - 2}$

Step 8.2.3

Rewrite the expression.

$- \frac{1.20710678 (- 2 + 2 \sqrt{2})}{- 2}$

Step 9

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

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Step 9.1

Factor $2$ out of $1.20710678 (- 2 + 2 \sqrt{2})$ .

$- \frac{2 (0.60355339 (- 2 + 2 \sqrt{2}))}{- 2}$

Step 9.2

Move the negative one from the denominator of $\frac{0.60355339 (- 2 + 2 \sqrt{2})}{- 1}$ .

$- (- 1 \cdot (0.60355339 (- 2 + 2 \sqrt{2})))$

Step 10

Rewrite $- 1 \cdot (0.60355339 (- 2 + 2 \sqrt{2}))$ as $- (0.60355339 (- 2 + 2 \sqrt{2}))$ .

$- - (0.60355339 (- 2 + 2 \sqrt{2}))$

Step 11

Multiply $0.60355339$ by $- 2 + 2 \sqrt{2}$ .

$- (- 1 \cdot 0.5)$

Step 12

Multiply $- (- 1 \cdot 0.5)$ .

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Step 12.1

Multiply $- 1$ by $0.5$ .

$- - 0.5$

Step 12.2

Multiply $- 1$ by $- 0.5$ .

$0.5$