Find the Exact Value arctan(tan(-pi/8))

$\arctan (\tan (- \frac{π}{8}))$

Step 1

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

$\arctan (\tan (\frac{15 π}{8}))$

Step 2

The exact value of $\tan (\frac{15 π}{8})$ is $- \sqrt{3 - 2 \sqrt{2}}$ .

Rewrite $\frac{15 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\arctan (\tan (\frac{\frac{15 π}{4}}{2}))$

Apply the tangent half-angle identity.

$\arctan (\pm \sqrt{\frac{1 - \cos (\frac{15 π}{4})}{1 + \cos (\frac{15 π}{4})}})$

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

$\arctan (- \sqrt{\frac{1 - \cos (\frac{15 π}{4})}{1 + \cos (\frac{15 π}{4})}})$

Simplify $- \sqrt{\frac{1 - \cos (\frac{15 π}{4})}{1 + \cos (\frac{15 π}{4})}}$ .

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

$\arctan (- \sqrt{\frac{1 - \cos (\frac{7 π}{4})}{1 + \cos (\frac{15 π}{4})}})$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\arctan (- \sqrt{\frac{1 - \cos (\frac{π}{4})}{1 + \cos (\frac{15 π}{4})}})$

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

$\arctan (- \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \cos (\frac{15 π}{4})}})$

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

$\arctan (- \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{1 + \cos (\frac{15 π}{4})}})$

Combine the numerators over the common denominator.

$\arctan (- \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{1 + \cos (\frac{15 π}{4})}})$

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

$\arctan (- \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{1 + \cos (\frac{7 π}{4})}})$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Apply the distributive property.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify each term.

Apply the distributive property.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $2$ .

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

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

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

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

Factor $2$ out of $- 2$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Add $2$ and $1$ .

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

Subtract $\sqrt{2}$ from $- \sqrt{2}$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.39269908 \dots$

Find the Exact Value arctan(tan(-pi/8))

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