Trigonometry Examples

$\arctan (\tan (\frac{7 π}{12}))$

Step 1

Rewrite $\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\arctan (\tan (\frac{\frac{7 π}{6}}{2}))$

Step 2

Apply the tangent half-angle identity.

$\arctan (\pm \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}})$

Step 3

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

$\arctan (- \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}})$

Step 4

Simplify $- \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}}$ .

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Step 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.

$\arctan (- \sqrt{\frac{1 - - \cos (\frac{π}{6})}{1 + \cos (\frac{7 π}{6})}})$

Step 4.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\arctan (- \sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})$

Step 4.3

Multiply $- - \frac{\sqrt{3}}{2}$ .

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

$\arctan (- \sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})$

Step 4.3.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\arctan (- \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})$

Step 4.4

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

$\arctan (- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})$

Step 4.5

Combine the numerators over the common denominator.

$\arctan (- \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})$

Step 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.

$\arctan (- \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 - \cos (\frac{π}{6})}})$

Step 4.7

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 4.8

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

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

Step 4.9

Combine the numerators over the common denominator.

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

Step 4.10

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.11

Cancel the common factor of $2$ .

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Step 4.11.1

Cancel the common factor.

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

Step 4.11.2

Rewrite the expression.

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

Step 4.12

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

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

Step 4.13

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

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

Step 4.14

Expand the denominator using the FOIL method.

$\arctan (- \sqrt{(2 + \sqrt{3}) \frac{2 + \sqrt{3}}{4 + 2 \sqrt{3} - 2 \sqrt{3} - {\sqrt{3}}^{2}}})$

Step 4.15

Simplify.

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

Step 4.16

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

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

Step 4.17

Expand $(2 + \sqrt{3}) (2 + \sqrt{3})$ using the FOIL Method.

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Step 4.17.1

Apply the distributive property.

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

Step 4.17.2

Apply the distributive property.

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

Step 4.17.3

Apply the distributive property.

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

Step 4.18

Simplify and combine like terms.

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Step 4.18.1

Simplify each term.

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Step 4.18.1.1

Multiply $2$ by $2$ .

$\arctan (- \sqrt{4 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}})$

Step 4.18.1.2

Move $2$ to the left of $\sqrt{3}$ .

$\arctan (- \sqrt{4 + 2 \sqrt{3} + 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}})$

Step 4.18.1.3

Combine using the product rule for radicals.

$\arctan (- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3 \cdot 3}})$

Step 4.18.1.4

Multiply $3$ by $3$ .

$\arctan (- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{9}})$

Step 4.18.1.5

Rewrite $9$ as $3^{2}$ .

$\arctan (- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3^{2}}})$

Step 4.18.1.6

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

$\arctan (- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3})$

Step 4.18.2

Add $4$ and $3$ .

$\arctan (- \sqrt{7 + 2 \sqrt{3} + 2 \sqrt{3}})$

Step 4.18.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$\arctan (- \sqrt{7 + 4 \sqrt{3}})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\arctan (- \sqrt{7 + 4 \sqrt{3}})$

Decimal Form:

$- 1.30899693 \dots$