Trigonometry Examples

$\tan (\frac{x}{7}) + \sqrt{3} = 0$

Step 1

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

$\tan (\frac{x}{7}) = - \sqrt{3}$

Step 2

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

$\frac{x}{7} = \arctan (- \sqrt{3})$

Step 3

Simplify the right side.

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Step 3.1

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

$\frac{x}{7} = - \frac{π}{3}$

Step 4

Multiply both sides of the equation by $7$ .

$7 \frac{x}{7} = 7 (- \frac{π}{3})$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $7$ .

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Step 5.1.1.1

Cancel the common factor.

$7 \frac{x}{7} = 7 (- \frac{π}{3})$

Step 5.1.1.2

Rewrite the expression.

$x = 7 (- \frac{π}{3})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $7 (- \frac{π}{3})$ .

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Step 5.2.1.1

Multiply $7 (- \frac{π}{3})$ .

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Step 5.2.1.1.1

Multiply $- 1$ by $7$ .

$x = - 7 \frac{π}{3}$

Step 5.2.1.1.2

Combine $- 7$ and $\frac{π}{3}$ .

$x = \frac{- 7 π}{3}$

Step 5.2.1.2

Move the negative in front of the fraction.

$x = - \frac{7 π}{3}$

Step 6

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

$\frac{x}{7} = - \frac{π}{3} - π$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

Add $2 π$ to $- \frac{π}{3} - π$ .

$\frac{x}{7} = - \frac{π}{3} - π + 2 π$

Step 7.2

The resulting angle of $\frac{2 π}{3}$ is positive and coterminal with $- \frac{π}{3} - π$ .

$\frac{x}{7} = \frac{2 π}{3}$

Step 7.3

Solve for $x$ .

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Step 7.3.1

Multiply both sides of the equation by $7$ .

$7 \frac{x}{7} = 7 \frac{2 π}{3}$

Step 7.3.2

Simplify both sides of the equation.

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Step 7.3.2.1

Simplify the left side.

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Step 7.3.2.1.1

Cancel the common factor of $7$ .

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Step 7.3.2.1.1.1

Cancel the common factor.

$7 \frac{x}{7} = 7 \frac{2 π}{3}$

Step 7.3.2.1.1.2

Rewrite the expression.

$x = 7 \frac{2 π}{3}$

Step 7.3.2.2

Simplify the right side.

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Step 7.3.2.2.1

Multiply $7 \frac{2 π}{3}$ .

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Step 7.3.2.2.1.1

Combine $7$ and $\frac{2 π}{3}$ .

$x = \frac{7 (2 π)}{3}$

Step 7.3.2.2.1.2

Multiply $2$ by $7$ .

$x = \frac{14 π}{3}$

Step 8

Find the period of $\tan (\frac{x}{7})$ .

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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 $\frac{1}{7}$ in the formula for period.

$\frac{π}{| \frac{1}{7} |}$

Step 8.3

$\frac{1}{7}$ is approximately $0.\overline{142857}$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{7}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 7$

Step 8.5

Move $7$ to the left of $π$ .

$7 π$

Step 9

Add $7 π$ to every negative angle to get positive angles.

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Step 9.1

Add $7 π$ to $- \frac{7 π}{3}$ to find the positive angle.

$- \frac{7 π}{3} + 7 π$

Step 9.2

To write $7 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$7 π \cdot \frac{3}{3} - \frac{7 π}{3}$

Step 9.3

Combine fractions.

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Step 9.3.1

Combine $7 π$ and $\frac{3}{3}$ .

$\frac{7 π \cdot 3}{3} - \frac{7 π}{3}$

Step 9.3.2

Combine the numerators over the common denominator.

$\frac{7 π \cdot 3 - 7 π}{3}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

Multiply $3$ by $7$ .

$\frac{21 π - 7 π}{3}$

Step 9.4.2

Subtract $7 π$ from $21 π$ .

$\frac{14 π}{3}$

Step 9.5

List the new angles.

$x = \frac{14 π}{3}$

Step 10

The period of the $\tan (\frac{x}{7})$ function is $7 π$ so values will repeat every $7 π$ radians in both directions.

$x = \frac{14 π}{3} + 7 π n, \frac{14 π}{3} + 7 π n$ , for any integer $n$

Step 11

Consolidate the answers.

$x = \frac{14 π}{3} + 7 π n$ , for any integer $n$