Calculus Examples

$\tan (θ) = - \frac{2 \sqrt{3}}{3} \sin (θ)$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $3$ .

$\tan (θ) \cdot \frac{3}{3} = - \frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $3$ .

$\tan (θ) \cdot 3$

Step 3

Move $3$ to the left of $\tan (θ)$ .

$\frac{3 \tan (θ)}{3} = - \frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$

Step 4

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $3$ .

$\sin (θ) \cdot 3$

Step 5

Move $3$ to the left of $\sin (θ)$ .

$\frac{3 \tan (θ)}{3} = - \frac{2 \sqrt{3}}{3} \cdot \frac{3 \sin (θ)}{3}$

Step 6

Simplify the left side.

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Step 6.1

Simplify $\tan (θ) \cdot \frac{3}{3}$ .

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Step 6.1.1

Divide $3$ by $3$ .

$\tan (θ) \cdot 1 = - \frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$

Step 6.1.2

Multiply $\tan (θ)$ by $1$ .

$\tan (θ) = - \frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$

Step 7

Simplify the right side.

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Step 7.1

Simplify $- \frac{2 \sqrt{3}}{3} \cdot \sin (θ) \cdot \frac{3}{3}$ .

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Step 7.1.1

Combine $\sin (θ)$ and $\frac{2 \sqrt{3}}{3}$ .

$\tan (θ) = - \frac{\sin (θ) (2 \sqrt{3})}{3} \cdot \frac{3}{3}$

Step 7.1.2

Cancel the common factor of $3$ .

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Step 7.1.2.1

Cancel the common factor.

$\tan (θ) = - \frac{\sin (θ) (2 \sqrt{3})}{3} \cdot \frac{3}{3}$

Step 7.1.2.2

Rewrite the expression.

$\tan (θ) = - \frac{\sin (θ) (2 \sqrt{3})}{3} \cdot 1$

Step 7.1.3

Simplify the expression.

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Step 7.1.3.1

Multiply $- \frac{\sin (θ) (2 \sqrt{3})}{3}$ by $1$ .

$\tan (θ) = - \frac{\sin (θ) (2 \sqrt{3})}{3}$

Step 7.1.3.2

Move $2$ to the left of $\sin (θ)$ .

$\tan (θ) = - \frac{2 \sin (θ) \sqrt{3}}{3}$

Step 8

Divide each term in $\tan (θ) = - \frac{2 \sin (θ) \sqrt{3}}{3}$ by $\tan (θ)$ and simplify.

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Step 8.1

Divide each term in $\tan (θ) = - \frac{2 \sin (θ) \sqrt{3}}{3}$ by $\tan (θ)$ .

$\frac{\tan (θ)}{\tan (θ)} = \frac{- \frac{2 \sin (θ) \sqrt{3}}{3}}{\tan (θ)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\tan (θ)$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{\tan (θ)}{\tan (θ)} = \frac{- \frac{2 \sin (θ) \sqrt{3}}{3}}{\tan (θ)}$

Step 8.2.1.2

Rewrite the expression.

$1 = \frac{- \frac{2 \sin (θ) \sqrt{3}}{3}}{\tan (θ)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

$1 = - \frac{2 \sin (θ) \sqrt{3}}{3} \cdot \frac{1}{\tan (θ)}$

Step 8.3.2

Rewrite $\tan (θ)$ in terms of sines and cosines.

$1 = - \frac{2 \sin (θ) \sqrt{3}}{3} \cdot \frac{1}{\frac{\sin (θ)}{\cos (θ)}}$

Step 8.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (θ)}{\cos (θ)}$ .

$1 = - \frac{2 \sin (θ) \sqrt{3}}{3} (1 \frac{\cos (θ)}{\sin (θ)})$

Step 8.3.4

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

$1 = - \frac{2 \sin (θ) \sqrt{3}}{3} (\frac{1}{1} \cdot \frac{\cos (θ)}{\sin (θ)})$

Step 8.3.5

Simplify.

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Step 8.3.5.1

Rewrite the expression.

$1 = - \frac{2 \sin (θ) \sqrt{3}}{3} (1 \frac{\cos (θ)}{\sin (θ)})$

Step 8.3.5.2

Multiply $\frac{\cos (θ)}{\sin (θ)}$ by $1$ .

$1 = - \frac{2 \sin (θ) \sqrt{3}}{3} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 8.3.6

Cancel the common factor of $\sin (θ)$ .

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Step 8.3.6.1

Move the leading negative in $- \frac{2 \sin (θ) \sqrt{3}}{3}$ into the numerator.

$1 = \frac{- 2 \sin (θ) \sqrt{3}}{3} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 8.3.6.2

Factor $\sin (θ)$ out of $- 2 \sin (θ) \sqrt{3}$ .

$1 = \frac{\sin (θ) (- 2 \sqrt{3})}{3} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 8.3.6.3

Cancel the common factor.

$1 = \frac{\sin (θ) (- 2 \sqrt{3})}{3} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 8.3.6.4

Rewrite the expression.

$1 = \frac{- 2 \sqrt{3}}{3} \cos (θ)$

Step 8.3.7

Combine $\frac{- 2 \sqrt{3}}{3}$ and $\cos (θ)$ .

$1 = \frac{- 2 \sqrt{3} \cos (θ)}{3}$

Step 8.3.8

Move the negative in front of the fraction.

$1 = - \frac{2 \sqrt{3} \cos (θ)}{3}$

Step 9

Rewrite the equation as $- \frac{2 \sqrt{3} \cos (θ)}{3} = 1$ .

$- \frac{2 \sqrt{3} \cos (θ)}{3} = 1$

Step 10

Multiply both sides of the equation by $- \frac{3}{2 \sqrt{3}}$ .

$- \frac{3}{2 \sqrt{3}} (- \frac{2 \sqrt{3} \cos (θ)}{3}) = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11

Simplify both sides of the equation.

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Step 11.1

Simplify the left side.

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Step 11.1.1

Simplify $- \frac{3}{2 \sqrt{3}} (- \frac{2 \sqrt{3} \cos (θ)}{3})$ .

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Step 11.1.1.1

Cancel the common factor of $3$ .

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Step 11.1.1.1.1

Move the leading negative in $- \frac{3}{2 \sqrt{3}}$ into the numerator.

$\frac{- 3}{2 \sqrt{3}} (- \frac{2 \sqrt{3} \cos (θ)}{3}) = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.1.1.1.2

Move the leading negative in $- \frac{2 \sqrt{3} \cos (θ)}{3}$ into the numerator.

$\frac{- 3}{2 \sqrt{3}} \cdot \frac{- 2 \sqrt{3} \cos (θ)}{3} = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2 \sqrt{3}} \cdot \frac{- 2 \sqrt{3} \cos (θ)}{3} = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2 \sqrt{3}} \cdot \frac{- 2 \sqrt{3} \cos (θ)}{3} = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2 \sqrt{3}} (- 2 \sqrt{3} \cos (θ)) = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.1.1.2

Cancel the common factor of $2 \sqrt{3}$ .

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Step 11.1.1.2.1

Factor $2 \sqrt{3}$ out of $- 2 \sqrt{3} \cos (θ)$ .

$\frac{- 1}{2 \sqrt{3}} (2 \sqrt{3} (- 1 \cos (θ))) = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2 \sqrt{3}} (2 \sqrt{3} (- 1 \cos (θ))) = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.1.1.2.3

Rewrite the expression.

$- (- 1 \cos (θ)) = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.1.1.3

Multiply.

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Step 11.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \cos (θ) = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.1.1.3.2

Multiply $\cos (θ)$ by $1$ .

$\cos (θ) = - \frac{3}{2 \sqrt{3}} \cdot 1$

Step 11.2

Simplify the right side.

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Step 11.2.1

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

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Step 11.2.1.1

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

$\cos (θ) = - (\frac{3}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) \cdot 1$

Step 11.2.1.2

Combine and simplify the denominator.

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Step 11.2.1.2.1

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

$\cos (θ) = - \frac{3 \sqrt{3}}{2 \sqrt{3} \sqrt{3}} \cdot 1$

Step 11.2.1.2.2

Move $\sqrt{3}$ .

$\cos (θ) = - \frac{3 \sqrt{3}}{2 (\sqrt{3} \sqrt{3})} \cdot 1$

Step 11.2.1.2.3

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

$\cos (θ) = - \frac{3 \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})} \cdot 1$

Step 11.2.1.2.4

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

$\cos (θ) = - \frac{3 \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})} \cdot 1$

Step 11.2.1.2.5

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

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

Step 11.2.1.2.6

Add $1$ and $1$ .

$\cos (θ) = - \frac{3 \sqrt{3}}{2 {\sqrt{3}}^{2}} \cdot 1$

Step 11.2.1.2.7

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

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Step 11.2.1.2.7.1

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

$\cos (θ) = - \frac{3 \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}} \cdot 1$

Step 11.2.1.2.7.2

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

$\cos (θ) = - \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{1}{2} \cdot 2}} \cdot 1$

Step 11.2.1.2.7.3

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

$\cos (θ) = - \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} \cdot 1$

Step 11.2.1.2.7.4

Cancel the common factor of $2$ .

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Step 11.2.1.2.7.4.1

Cancel the common factor.

$\cos (θ) = - \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}} \cdot 1$

Step 11.2.1.2.7.4.2

Rewrite the expression.

$\cos (θ) = - \frac{3 \sqrt{3}}{2 \cdot 3^{1}} \cdot 1$

Step 11.2.1.2.7.5

Evaluate the exponent.

$\cos (θ) = - \frac{3 \sqrt{3}}{2 \cdot 3} \cdot 1$

Step 11.2.1.3

Multiply $2$ by $3$ .

$\cos (θ) = - \frac{3 \sqrt{3}}{6} \cdot 1$

Step 11.2.1.4

Cancel the common factor of $3$ and $6$ .

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Step 11.2.1.4.1

Factor $3$ out of $3 \sqrt{3}$ .

$\cos (θ) = - \frac{3 (\sqrt{3})}{6} \cdot 1$

Step 11.2.1.4.2

Cancel the common factors.

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Step 11.2.1.4.2.1

Factor $3$ out of $6$ .

$\cos (θ) = - \frac{3 \sqrt{3}}{3 \cdot 2} \cdot 1$

Step 11.2.1.4.2.2

Cancel the common factor.

$\cos (θ) = - \frac{3 \sqrt{3}}{3 \cdot 2} \cdot 1$

Step 11.2.1.4.2.3

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{3}}{2} \cdot 1$

Step 11.2.1.5

Multiply $- 1$ by $1$ .

$\cos (θ) = - \frac{\sqrt{3}}{2}$

Step 12

Take the inverse cosine of both sides of the equation to extract $θ$ from inside the cosine.

$θ = \arccos (- \frac{\sqrt{3}}{2})$

Step 13

Simplify the right side.

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Step 13.1

The exact value of $\arccos (- \frac{\sqrt{3}}{2})$ is $\frac{5 π}{6}$ .

$θ = \frac{5 π}{6}$

Step 14

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$θ = 2 π - \frac{5 π}{6}$

Step 15

Simplify $2 π - \frac{5 π}{6}$ .

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Step 15.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$θ = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 15.2

Combine fractions.

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Step 15.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$θ = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 15.2.2

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 6 - 5 π}{6}$

Step 15.3

Simplify the numerator.

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Step 15.3.1

Multiply $6$ by $2$ .

$θ = \frac{12 π - 5 π}{6}$

Step 15.3.2

Subtract $5 π$ from $12 π$ .

$θ = \frac{7 π}{6}$

Step 16

Find the period of $\cos (θ)$ .

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Step 16.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 16.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 16.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 16.4

Divide $2 π$ by $1$ .

$2 π$

Step 17

The period of the $\cos (θ)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$θ = \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$