Trigonometry Examples

$\sqrt{3} - \tan (3 θ) = 0$

Step 1

Subtract $\sqrt{3}$ from both sides of the equation.

$- \tan (3 θ) = - \sqrt{3}$

Step 2

Divide each term in $- \tan (3 θ) = - \sqrt{3}$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \tan (3 θ) = - \sqrt{3}$ by $- 1$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2

Divide $\tan (3 θ)$ by $1$ .

$\tan (3 θ) = \frac{- \sqrt{3}}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Dividing two negative values results in a positive value.

$\tan (3 θ) = \frac{\sqrt{3}}{1}$

Step 2.3.2

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

$\tan (3 θ) = \sqrt{3}$

Step 3

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

$3 θ = \arctan (\sqrt{3})$

Step 4

Simplify the right side.

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Step 4.1

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

$3 θ = \frac{π}{3}$

Step 5

Divide each term in $3 θ = \frac{π}{3}$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 θ = \frac{π}{3}$ by $3$ .

$\frac{3 θ}{3} = \frac{\frac{π}{3}}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 θ}{3} = \frac{\frac{π}{3}}{3}$

Step 5.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{3}}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{π}{3} \cdot \frac{1}{3}$

Step 5.3.2

Multiply $\frac{π}{3} \cdot \frac{1}{3}$ .

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Step 5.3.2.1

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

$θ = \frac{π}{3 \cdot 3}$

Step 5.3.2.2

Multiply $3$ by $3$ .

$θ = \frac{π}{9}$

Step 6

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$3 θ = π + \frac{π}{3}$

Step 7

Solve for $θ$ .

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Step 7.1

Simplify.

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Step 7.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 θ = π \cdot \frac{3}{3} + \frac{π}{3}$

Step 7.1.2

Combine $π$ and $\frac{3}{3}$ .

$3 θ = \frac{π \cdot 3}{3} + \frac{π}{3}$

Step 7.1.3

Combine the numerators over the common denominator.

$3 θ = \frac{π \cdot 3 + π}{3}$

Step 7.1.4

Add $π \cdot 3$ and $π$ .

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Step 7.1.4.1

Reorder $π$ and $3$ .

$3 θ = \frac{3 \cdot π + π}{3}$

Step 7.1.4.2

Add $3 \cdot π$ and $π$ .

$3 θ = \frac{4 \cdot π}{3}$

Step 7.2

Divide each term in $3 θ = \frac{4 \cdot π}{3}$ by $3$ and simplify.

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Step 7.2.1

Divide each term in $3 θ = \frac{4 \cdot π}{3}$ by $3$ .

$\frac{3 θ}{3} = \frac{\frac{4 \cdot π}{3}}{3}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{3 θ}{3} = \frac{\frac{4 \cdot π}{3}}{3}$

Step 7.2.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{4 \cdot π}{3}}{3}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{4 \cdot π}{3} \cdot \frac{1}{3}$

Step 7.2.3.2

Multiply $\frac{4 π}{3} \cdot \frac{1}{3}$ .

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Step 7.2.3.2.1

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

$θ = \frac{4 π}{3 \cdot 3}$

Step 7.2.3.2.2

Multiply $3$ by $3$ .

$θ = \frac{4 π}{9}$

Step 8

Find the period of $\tan (3 θ)$ .

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

Replace $b$ with $3$ in the formula for period.

$\frac{π}{| 3 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{π}{3}$

Step 9

The period of the $\tan (3 θ)$ function is $\frac{π}{3}$ so values will repeat every $\frac{π}{3}$ radians in both directions.

$θ = \frac{π}{9} + \frac{π n}{3}, \frac{4 π}{9} + \frac{π n}{3}$ , for any integer $n$

Step 10

Consolidate the answers.

$θ = \frac{π}{9} + \frac{π n}{3}$ , for any integer $n$