Trigonometry Examples

$\arctan (\tan (- \frac{3 π}{8}))$

Step 1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\arctan (\tan (\frac{13 π}{8}))$

Step 2

The exact value of $\tan (\frac{13 π}{8})$ is $- \sqrt{3 + 2 \sqrt{2}}$ .

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Step 2.1

Rewrite $\frac{13 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\arctan (\tan (\frac{\frac{13 π}{4}}{2}))$

Step 2.2

Apply the tangent half-angle identity.

$\arctan (\pm \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{1 + \cos (\frac{13 π}{4})}})$

Step 2.3

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

$\arctan (- \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{1 + \cos (\frac{13 π}{4})}})$

Step 2.4

Simplify $- \sqrt{\frac{1 - \cos (\frac{13 π}{4})}{1 + \cos (\frac{13 π}{4})}}$ .

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Step 2.4.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\arctan (- \sqrt{\frac{1 - \cos (\frac{5 π}{4})}{1 + \cos (\frac{13 π}{4})}})$

Step 2.4.2

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.

$\arctan (- \sqrt{\frac{1 - - \cos (\frac{π}{4})}{1 + \cos (\frac{13 π}{4})}})$

Step 2.4.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\arctan (- \sqrt{\frac{1 - - \frac{\sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}})$

Step 2.4.4

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

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Step 2.4.4.1

Multiply $- 1$ by $- 1$ .

$\arctan (- \sqrt{\frac{1 + 1 \frac{\sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}})$

Step 2.4.4.2

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

$\arctan (- \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}})$

Step 2.4.5

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

$\arctan (- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}})$

Step 2.4.6

Combine the numerators over the common denominator.

$\arctan (- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 + \cos (\frac{13 π}{4})}})$

Step 2.4.7

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\arctan (- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 + \cos (\frac{5 π}{4})}})$

Step 2.4.8

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.

$\arctan (- \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{1 - \cos (\frac{π}{4})}})$

Step 2.4.9

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 2.4.10

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

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

Step 2.4.11

Combine the numerators over the common denominator.

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

Step 2.4.12

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.13

Cancel the common factor of $2$ .

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Step 2.4.13.1

Cancel the common factor.

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

Step 2.4.13.2

Rewrite the expression.

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

Step 2.4.14

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

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

Step 2.4.15

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

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

Step 2.4.16

Expand the denominator using the FOIL method.

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

Step 2.4.17

Simplify.

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

Step 2.4.18

Apply the distributive property.

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

Step 2.4.19

Cancel the common factor of $2$ .

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Step 2.4.19.1

Cancel the common factor.

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

Step 2.4.19.2

Rewrite the expression.

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

Step 2.4.20

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

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

Step 2.4.21

Simplify each term.

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Step 2.4.21.1

Apply the distributive property.

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

Step 2.4.21.2

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

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

Step 2.4.21.3

Combine using the product rule for radicals.

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

Step 2.4.21.4

Simplify each term.

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Step 2.4.21.4.1

Multiply $2$ by $2$ .

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

Step 2.4.21.4.2

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

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

Step 2.4.21.4.3

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

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

Step 2.4.21.5

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

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Step 2.4.21.5.1

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

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

Step 2.4.21.5.2

Factor $2$ out of $2$ .

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

Step 2.4.21.5.3

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

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

Step 2.4.21.5.4

Cancel the common factors.

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Step 2.4.21.5.4.1

Factor $2$ out of $2$ .

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

Step 2.4.21.5.4.2

Cancel the common factor.

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

Step 2.4.21.5.4.3

Rewrite the expression.

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

Step 2.4.21.5.4.4

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

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

Step 2.4.22

Add $2$ and $1$ .

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

Step 2.4.23

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.17809724 \dots$