Trigonometry Examples

$5 \cos (x) = 5 \sqrt{3} \sin (- x)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $5 \sqrt{3} \sin (- x)$ .

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Step 1.1.1

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$5 \cos (x) = 5 \sqrt{3} (- \sin (x))$

Step 1.1.2

Multiply $- 1$ by $5$ .

$5 \cos (x) = - 5 \sqrt{3} \sin (x)$

Step 2

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

$\frac{5 \cos (x)}{\cos (x)} = \frac{- 5 \sqrt{3} \sin (x)}{\cos (x)}$

Step 3

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

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Step 3.1

Cancel the common factor.

$\frac{5 \cos (x)}{\cos (x)} = \frac{- 5 \sqrt{3} \sin (x)}{\cos (x)}$

Step 3.2

Divide $5$ by $1$ .

$5 = \frac{- 5 \sqrt{3} \sin (x)}{\cos (x)}$

Step 4

Separate fractions.

$5 = \frac{- 5 \sqrt{3}}{1} \cdot \frac{\sin (x)}{\cos (x)}$

Step 5

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

$5 = \frac{- 5 \sqrt{3}}{1} \cdot \tan (x)$

Step 6

Divide $- 5 \sqrt{3}$ by $1$ .

$5 = - 5 \sqrt{3} \tan (x)$

Step 7

Rewrite the equation as $- 5 \sqrt{3} \tan (x) = 5$ .

$- 5 \sqrt{3} \tan (x) = 5$

Step 8

Divide each term in $- 5 \sqrt{3} \tan (x) = 5$ by $- 5 \sqrt{3}$ and simplify.

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Step 8.1

Divide each term in $- 5 \sqrt{3} \tan (x) = 5$ by $- 5 \sqrt{3}$ .

$\frac{- 5 \sqrt{3} \tan (x)}{- 5 \sqrt{3}} = \frac{5}{- 5 \sqrt{3}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $- 5$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{- 5 \sqrt{3} \tan (x)}{- 5 \sqrt{3}} = \frac{5}{- 5 \sqrt{3}}$

Step 8.2.1.2

Rewrite the expression.

$\frac{\sqrt{3} \tan (x)}{\sqrt{3}} = \frac{5}{- 5 \sqrt{3}}$

Step 8.2.2

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

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Step 8.2.2.1

Cancel the common factor.

$\frac{\sqrt{3} \tan (x)}{\sqrt{3}} = \frac{5}{- 5 \sqrt{3}}$

Step 8.2.2.2

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

$\tan (x) = \frac{5}{- 5 \sqrt{3}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $5$ and $- 5$ .

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Step 8.3.1.1

Factor $5$ out of $5$ .

$\tan (x) = \frac{5 (1)}{- 5 \sqrt{3}}$

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Factor $5$ out of $- 5 \sqrt{3}$ .

$\tan (x) = \frac{5 (1)}{5 (- \sqrt{3})}$

Step 8.3.1.2.2

Cancel the common factor.

$\tan (x) = \frac{5 \cdot 1}{5 (- \sqrt{3})}$

Step 8.3.1.2.3

Rewrite the expression.

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

Step 8.3.2

Cancel the common factor of $1$ and $- 1$ .

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Step 8.3.2.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 8.3.2.2

Move the negative in front of the fraction.

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

Step 8.3.3

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

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

Step 8.3.4

Combine and simplify the denominator.

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Step 8.3.4.1

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

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

Step 8.3.4.2

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

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

Step 8.3.4.3

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

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

Step 8.3.4.4

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

$\tan (x) = - \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 8.3.4.5

Add $1$ and $1$ .

$\tan (x) = - \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 8.3.4.6

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

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Step 8.3.4.6.1

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

$\tan (x) = - \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 8.3.4.6.2

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

$\tan (x) = - \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 8.3.4.6.3

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

$\tan (x) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.4.6.4

Cancel the common factor of $2$ .

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Step 8.3.4.6.4.1

Cancel the common factor.

$\tan (x) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.4.6.4.2

Rewrite the expression.

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

Step 8.3.4.6.5

Evaluate the exponent.

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

Step 9

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

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

Step 10

Simplify the right side.

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Step 10.1

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

$x = - \frac{π}{6}$

Step 11

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 12

Simplify the expression to find the second solution.

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Step 12.1

Add $2 π$ to $- \frac{π}{6} - π$ .

$x = - \frac{π}{6} - π + 2 π$

Step 12.2

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

$x = \frac{5 π}{6}$

Step 13

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

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Step 13.1

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

$\frac{π}{| b |}$

Step 13.2

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

$\frac{π}{| 1 |}$

Step 13.3

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

$\frac{π}{1}$

Step 13.4

Divide $π$ by $1$ .

$π$

Step 14

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

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Step 14.1

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

$- \frac{π}{6} + π$

Step 14.2

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

$π \cdot \frac{6}{6} - \frac{π}{6}$

Step 14.3

Combine fractions.

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Step 14.3.1

Combine $π$ and $\frac{6}{6}$ .

$\frac{π \cdot 6}{6} - \frac{π}{6}$

Step 14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 6 - π}{6}$

Step 14.4

Simplify the numerator.

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Step 14.4.1

Move $6$ to the left of $π$ .

$\frac{6 \cdot π - π}{6}$

Step 14.4.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{6}$

Step 14.5

List the new angles.

$x = \frac{5 π}{6}$

Step 15

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 16

Consolidate the answers.

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