Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1.1

Add parentheses.

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

Step 1.1.2

Rewrite $3 \tan (x) \cos (x)$ in terms of sines and cosines.

$3 (\frac{\sin (x)}{\cos (x)} \cos (x)) = - \sqrt{3} \cos (x)$

Step 1.1.3

Cancel the common factors.

$3 \sin (x) = - \sqrt{3} \cos (x)$

Step 2

Divide each term in the equation by $\cos (x)$ .

$\frac{3 \sin (x)}{\cos (x)} = \frac{- \sqrt{3} \cos (x)}{\cos (x)}$

Step 3

Separate fractions.

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

Step 4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{3}{1} \cdot \tan (x) = \frac{- \sqrt{3} \cos (x)}{\cos (x)}$

Step 5

Divide $3$ by $1$ .

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

Step 6

Cancel the common factor of $\cos (x)$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $\tan (x)$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

Simplify the right side.

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Step 9.1

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

$x = - \frac{π}{6}$

Step 10

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.

$x = - \frac{π}{6} - π$

Step 11

Simplify the expression to find the second solution.

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Step 11.1

Add $2 π$ to $- \frac{π}{6} - π$ .

$x = - \frac{π}{6} - π + 2 π$

Step 11.2

The resulting angle of $\frac{5 π}{6}$ is positive and coterminal with $- \frac{π}{6} - π$ .

$x = \frac{5 π}{6}$

Step 12

Find the period of $\tan (x)$ .

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Step 12.1

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

$\frac{π}{| b |}$

Step 12.2

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

$\frac{π}{| 1 |}$

Step 12.3

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

$\frac{π}{1}$

Step 12.4

Divide $π$ by $1$ .

$π$

Step 13

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

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Step 13.1

Add $π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + π$

Step 13.2

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

$π \cdot \frac{6}{6} - \frac{π}{6}$

Step 13.3

Combine fractions.

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Step 13.3.1

Combine $π$ and $\frac{6}{6}$ .

$\frac{π \cdot 6}{6} - \frac{π}{6}$

Step 13.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 6 - π}{6}$

Step 13.4

Simplify the numerator.

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Step 13.4.1

Move $6$ to the left of $π$ .

$\frac{6 \cdot π - π}{6}$

Step 13.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{6}$

Step 13.5

List the new angles.

$x = \frac{5 π}{6}$

Step 14

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

$x = \frac{5 π}{6} + π n, \frac{5 π}{6} + π n$ , for any integer $n$

Step 15

Consolidate the answers.

$x = \frac{5 π}{6} + π n$ , for any integer $n$