Trigonometry Examples

$\cot (\frac{10 π}{3}) + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 1

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

$\cot (\frac{4 π}{3}) + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 2

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

$\cot (\frac{π}{3}) + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 3

The exact value of $\cot (\frac{π}{3})$ is $\frac{1}{\sqrt{3}}$ .

$\frac{1}{\sqrt{3}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 4

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

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5

Combine and simplify the denominator.

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Step 5.1

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

$\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.4

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.5

Add $1$ and $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{2}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.6

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

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Step 5.6.1

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

$\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.6.2

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

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.6.3

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

$\frac{\sqrt{3}}{3^{\frac{2}{2}}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$\frac{\sqrt{3}}{3^{\frac{2}{2}}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.6.4.2

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 5.6.5

Evaluate the exponent.

$\frac{\sqrt{3}}{3} + \frac{\cot (\frac{16 π}{3})}{\cos (\frac{17 π}{6})}$

Step 6

Rewrite $\cot (\frac{16 π}{3})$ in terms of sines and cosines.

$\frac{\sqrt{3}}{3} + \frac{\frac{\cos (\frac{16 π}{3})}{\sin (\frac{16 π}{3})}}{\cos (\frac{17 π}{6})}$

Step 7

Rewrite $\frac{\frac{\cos (\frac{16 π}{3})}{\sin (\frac{16 π}{3})}}{\cos (\frac{17 π}{6})}$ as a product.

$\frac{\sqrt{3}}{3} + \frac{\cos (\frac{16 π}{3})}{\sin (\frac{16 π}{3})} \cdot \frac{1}{\cos (\frac{17 π}{6})}$

Step 8

Multiply $\frac{\cos (\frac{16 π}{3})}{\sin (\frac{16 π}{3})}$ by $\frac{1}{\cos (\frac{17 π}{6})}$ .

$\frac{\sqrt{3}}{3} + \frac{\cos (\frac{16 π}{3})}{\sin (\frac{16 π}{3}) \cos (\frac{17 π}{6})}$

Step 9

Simplify the numerator.

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Step 9.1

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

$\frac{\sqrt{3}}{3} + \frac{\cos (\frac{4 π}{3})}{\sin (\frac{16 π}{3}) \cos (\frac{17 π}{6})}$

Step 9.2

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.

$\frac{\sqrt{3}}{3} + \frac{- \cos (\frac{π}{3})}{\sin (\frac{16 π}{3}) \cos (\frac{17 π}{6})}$

Step 9.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{\sqrt{3}}{3} + \frac{- \frac{1}{2}}{\sin (\frac{16 π}{3}) \cos (\frac{17 π}{6})}$

Step 10

Simplify the denominator.

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Step 10.1

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

$\frac{\sqrt{3}}{3} + \frac{- \frac{1}{2}}{\sin (\frac{4 π}{3}) \cos (\frac{17 π}{6})}$

Step 10.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$\frac{\sqrt{3}}{3} + \frac{- \frac{1}{2}}{- \sin (\frac{π}{3}) \cos (\frac{17 π}{6})}$

Step 10.3

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

$\frac{\sqrt{3}}{3} + \frac{- \frac{1}{2}}{- \frac{\sqrt{3}}{2} \cos (\frac{17 π}{6})}$

Step 10.4

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

$\frac{\sqrt{3}}{3} + \frac{- \frac{1}{2}}{- \frac{\sqrt{3}}{2} \cos (\frac{5 π}{6})}$

Step 10.5

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 second quadrant.

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

Step 10.6

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

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

Step 10.7

Combine exponents.

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Step 10.7.1

Factor out negative.

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

Step 10.7.2

Combine $\frac{\sqrt{3}}{2}$ and $- 1$ .

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

Step 10.7.3

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

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

Step 10.7.4

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 10.7.5

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 10.7.6

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

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

Step 10.7.7

Add $1$ and $1$ .

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

Step 10.7.8

Multiply $2$ by $2$ .

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

Step 10.8

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

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Step 10.8.1

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

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

Step 10.8.2

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

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

Step 10.8.3

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

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

Step 10.8.4

Cancel the common factor of $2$ .

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Step 10.8.4.1

Cancel the common factor.

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

Step 10.8.4.2

Rewrite the expression.

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

Step 10.8.5

Evaluate the exponent.

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

Step 10.9

Multiply $- 1$ by $3$ .

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

Step 10.10

Move the negative in front of the fraction.

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

Step 11

Simplify the denominator.

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Step 11.1

Multiply $- 1$ by $- 1$ .

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

Step 11.2

Multiply $\frac{3}{4}$ by $1$ .

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

Step 12

Multiply the numerator by the reciprocal of the denominator.

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

Step 13

Cancel the common factor of $2$ .

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Step 13.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 13.2

Factor $2$ out of $4$ .

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

Step 13.3

Cancel the common factor.

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

Step 13.4

Rewrite the expression.

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

Step 14

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.08931639 \dots$