Trigonometry Examples

$\cos (\frac{2 x}{3}) = - \frac{1}{2}$

Step 1

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

$\frac{2 x}{3} = \arccos (- \frac{1}{2})$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$\frac{2 x}{3} = \frac{2 π}{3}$

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 x = 2 π$

Step 4

Divide each term in $2 x = 2 π$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = 2 π$ by $2$ .

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Cancel the common factor.

$x = \frac{2 π}{2}$

Step 4.3.1.2

Divide $π$ by $1$ .

$x = π$

Step 5

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.

$\frac{2 x}{3} = 2 π - \frac{2 π}{3}$

Step 6

Solve for $x$ .

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Step 6.1

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 x}{3} = \frac{3}{2} (2 π - \frac{2 π}{3})$

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $\frac{3}{2} \cdot \frac{2 x}{3}$ .

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Step 6.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 x}{3} = \frac{3}{2} (2 π - \frac{2 π}{3})$

Step 6.2.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 x) = \frac{3}{2} (2 π - \frac{2 π}{3})$

Step 6.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.1.1.2.1

Factor $2$ out of $2 x$ .

$\frac{1}{2} (2 (x)) = \frac{3}{2} (2 π - \frac{2 π}{3})$

Step 6.2.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 x) = \frac{3}{2} (2 π - \frac{2 π}{3})$

Step 6.2.1.1.2.3

Rewrite the expression.

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

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\frac{3}{2} (2 π - \frac{2 π}{3})$ .

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Step 6.2.2.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{3}{2} (2 π \cdot \frac{3}{3} - \frac{2 π}{3})$

Step 6.2.2.1.2

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{3}{2} (\frac{2 π \cdot 3}{3} - \frac{2 π}{3})$

Step 6.2.2.1.3

Combine the numerators over the common denominator.

$x = \frac{3}{2} \cdot \frac{2 π \cdot 3 - 2 π}{3}$

Step 6.2.2.1.4

Cancel the common factor of $3$ .

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Step 6.2.2.1.4.1

Cancel the common factor.

$x = \frac{3}{2} \cdot \frac{2 π \cdot 3 - 2 π}{3}$

Step 6.2.2.1.4.2

Rewrite the expression.

$x = \frac{1}{2} (2 π \cdot 3 - 2 π)$

Step 6.2.2.1.5

Multiply $3$ by $2$ .

$x = \frac{1}{2} (6 π - 2 π)$

Step 6.2.2.1.6

Subtract $2 π$ from $6 π$ .

$x = \frac{1}{2} (4 π)$

Step 6.2.2.1.7

Cancel the common factor of $2$ .

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Step 6.2.2.1.7.1

Factor $2$ out of $4 π$ .

$x = \frac{1}{2} (2 (2 π))$

Step 6.2.2.1.7.2

Cancel the common factor.

$x = \frac{1}{2} (2 (2 π))$

Step 6.2.2.1.7.3

Rewrite the expression.

$x = 2 π$

Step 7

Find the period of $\cos (\frac{2 x}{3})$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $\frac{2}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2}{3} |}$

Step 7.3

$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{3}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2}$

Step 7.5

Cancel the common factor of $2$ .

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Step 7.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{2}$

Step 7.5.2

Cancel the common factor.

$2 π \frac{3}{2}$

Step 7.5.3

Rewrite the expression.

$π \cdot 3$

Step 7.6

Move $3$ to the left of $π$ .

$3 π$

Step 8

The period of the $\cos (\frac{2 x}{3})$ function is $3 π$ so values will repeat every $3 π$ radians in both directions.

$x = π + 3 π n, 2 π + 3 π n$ , for any integer $n$