Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the right side.

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Step 2.1

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

$- 2 x = \frac{5 π}{6}$

Step 3

Divide each term in $- 2 x = \frac{5 π}{6}$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 x = \frac{5 π}{6}$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{\frac{5 π}{6}}{- 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{\frac{5 π}{6}}{- 2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{6}}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 π}{6} \cdot \frac{1}{- 2}$

Step 3.3.2

Move the negative in front of the fraction.

$x = \frac{5 π}{6} (- \frac{1}{2})$

Step 3.3.3

Multiply $\frac{5 π}{6} (- \frac{1}{2})$ .

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Step 3.3.3.1

Multiply $\frac{5 π}{6}$ by $\frac{1}{2}$ .

$x = - \frac{5 π}{6 \cdot 2}$

Step 3.3.3.2

Multiply $6$ by $2$ .

$x = - \frac{5 π}{12}$

Step 4

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.

$- 2 x = 2 π - \frac{5 π}{6}$

Step 5

Solve for $x$ .

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Step 5.1

Simplify.

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Step 5.1.1

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

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

Step 5.1.2

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 5.1.3

Combine the numerators over the common denominator.

$- 2 x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 5.1.4

Multiply $6$ by $2$ .

$- 2 x = \frac{12 π - 5 π}{6}$

Step 5.1.5

Subtract $5 π$ from $12 π$ .

$- 2 x = \frac{7 π}{6}$

Step 5.2

Divide each term in $- 2 x = \frac{7 π}{6}$ by $- 2$ and simplify.

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Step 5.2.1

Divide each term in $- 2 x = \frac{7 π}{6}$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{\frac{7 π}{6}}{- 2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{\frac{7 π}{6}}{- 2}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 π}{6}}{- 2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{6} \cdot \frac{1}{- 2}$

Step 5.2.3.2

Move the negative in front of the fraction.

$x = \frac{7 π}{6} (- \frac{1}{2})$

Step 5.2.3.3

Multiply $\frac{7 π}{6} (- \frac{1}{2})$ .

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Step 5.2.3.3.1

Multiply $\frac{7 π}{6}$ by $\frac{1}{2}$ .

$x = - \frac{7 π}{6 \cdot 2}$

Step 5.2.3.3.2

Multiply $6$ by $2$ .

$x = - \frac{7 π}{12}$

Step 6

Find the period of $\cos (- 2 x)$ .

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

Replace $b$ with $- 2$ in the formula for period.

$\frac{2 π}{| - 2 |}$

Step 6.3

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$\frac{2 π}{2}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 6.4.2

Divide $π$ by $1$ .

$π$

Step 7

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

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Step 7.1

Add $π$ to $- \frac{5 π}{12}$ to find the positive angle.

$- \frac{5 π}{12} + π$

Step 7.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$π \cdot \frac{12}{12} - \frac{5 π}{12}$

Step 7.3

Combine fractions.

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Step 7.3.1

Combine $π$ and $\frac{12}{12}$ .

$\frac{π \cdot 12}{12} - \frac{5 π}{12}$

Step 7.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 12 - 5 π}{12}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Move $12$ to the left of $π$ .

$\frac{12 \cdot π - 5 π}{12}$

Step 7.4.2

Subtract $5 π$ from $12 π$ .

$\frac{7 π}{12}$

Step 7.5

Add $π$ to $- \frac{7 π}{12}$ to find the positive angle.

$- \frac{7 π}{12} + π$

Step 7.6

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$π \cdot \frac{12}{12} - \frac{7 π}{12}$

Step 7.7

Combine fractions.

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Step 7.7.1

Combine $π$ and $\frac{12}{12}$ .

$\frac{π \cdot 12}{12} - \frac{7 π}{12}$

Step 7.7.2

Combine the numerators over the common denominator.

$\frac{π \cdot 12 - 7 π}{12}$

Step 7.8

Simplify the numerator.

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Step 7.8.1

Move $12$ to the left of $π$ .

$\frac{12 \cdot π - 7 π}{12}$

Step 7.8.2

Subtract $7 π$ from $12 π$ .

$\frac{5 π}{12}$

Step 7.9

List the new angles.

$x = \frac{5 π}{12}$

Step 8

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

$x = \frac{5 π}{12} + π n, \frac{7 π}{12} + π n$ , for any integer $n$