Trigonometry Examples

$\cot (- \frac{5 π}{12})$

Step 1

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

$\cot (\frac{19 π}{12})$

Step 2

The exact value of $\cot (\frac{19 π}{12})$ is $- \frac{1}{\sqrt{7 + 4 \sqrt{3}}}$ .

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Step 2.1

Rewrite $\frac{19 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cot (\frac{\frac{19 π}{6}}{2})$

Step 2.2

Apply the reciprocal identity.

$\frac{1}{\tan (\frac{\frac{19 π}{6}}{2})}$

Step 2.3

Apply the tangent half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (\frac{19 π}{6})}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.4

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

$\frac{1}{- \sqrt{\frac{1 - \cos (\frac{19 π}{6})}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5

Simplify $\frac{1}{- \sqrt{\frac{1 - \cos (\frac{19 π}{6})}{1 + \cos (\frac{19 π}{6})}}}$ .

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Step 2.5.1

Cancel the common factor of $1$ and $- 1$ .

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Step 2.5.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \sqrt{\frac{1 - \cos (\frac{19 π}{6})}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5.1.2

Move the negative in front of the fraction.

$- \frac{1}{\sqrt{\frac{1 - \cos (\frac{19 π}{6})}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5.2

Simplify the numerator.

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Step 2.5.2.1

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

$- \frac{1}{\sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5.2.2

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.

$- \frac{1}{\sqrt{\frac{1 - - \cos (\frac{π}{6})}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5.2.3

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

$- \frac{1}{\sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5.2.4

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

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Step 2.5.2.4.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{\sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5.2.4.2

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

$- \frac{1}{\sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5.2.5

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

$- \frac{1}{\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5.2.6

Combine the numerators over the common denominator.

$- \frac{1}{\sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 + \cos (\frac{19 π}{6})}}}$

Step 2.5.3

Simplify the denominator.

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Step 2.5.3.1

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

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

Step 2.5.3.2

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.

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

Step 2.5.3.3

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

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

Step 2.5.3.4

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

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

Step 2.5.3.5

Combine the numerators over the common denominator.

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

Step 2.5.4

Simplify the denominator.

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Step 2.5.4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.4.2

Cancel the common factor of $2$ .

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Step 2.5.4.2.1

Cancel the common factor.

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

Step 2.5.4.2.2

Rewrite the expression.

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

Step 2.5.4.3

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

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

Step 2.5.4.4

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

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

Step 2.5.4.5

Expand the denominator using the FOIL method.

$- \frac{1}{\sqrt{(2 + \sqrt{3}) \frac{2 + \sqrt{3}}{4 + 2 \sqrt{3} - 2 \sqrt{3} - {\sqrt{3}}^{2}}}}$

Step 2.5.4.6

Simplify.

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

Step 2.5.4.7

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

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

Step 2.5.4.8

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

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Step 2.5.4.8.1

Apply the distributive property.

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

Step 2.5.4.8.2

Apply the distributive property.

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

Step 2.5.4.8.3

Apply the distributive property.

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

Step 2.5.4.9

Simplify and combine like terms.

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Step 2.5.4.9.1

Simplify each term.

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Step 2.5.4.9.1.1

Multiply $2$ by $2$ .

$- \frac{1}{\sqrt{4 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}}}$

Step 2.5.4.9.1.2

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

$- \frac{1}{\sqrt{4 + 2 \sqrt{3} + 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}}$

Step 2.5.4.9.1.3

Combine using the product rule for radicals.

$- \frac{1}{\sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3 \cdot 3}}}$

Step 2.5.4.9.1.4

Multiply $3$ by $3$ .

$- \frac{1}{\sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{9}}}$

Step 2.5.4.9.1.5

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

$- \frac{1}{\sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3^{2}}}}$

Step 2.5.4.9.1.6

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

$- \frac{1}{\sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3}}$

Step 2.5.4.9.2

Add $4$ and $3$ .

$- \frac{1}{\sqrt{7 + 2 \sqrt{3} + 2 \sqrt{3}}}$

Step 2.5.4.9.3

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

$- \frac{1}{\sqrt{7 + 4 \sqrt{3}}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{\sqrt{7 + 4 \sqrt{3}}}$

Decimal Form:

$- 0.26794919 \dots$