Trigonometry Examples

$2 \cos (3 x - 1) = 0$

Step 1

Divide each term in $2 \cos (3 x - 1) = 0$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \cos (3 x - 1) = 0$ by $2$ .

$\frac{2 \cos (3 x - 1)}{2} = \frac{0}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 \cos (3 x - 1)}{2} = \frac{0}{2}$

Step 1.2.1.2

Divide $\cos (3 x - 1)$ by $1$ .

$\cos (3 x - 1) = \frac{0}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $0$ by $2$ .

$\cos (3 x - 1) = 0$

Step 2

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

$3 x - 1 = \arccos (0)$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$3 x - 1 = \frac{π}{2}$

Step 4

Add $1$ to both sides of the equation.

$3 x = \frac{π}{2} + 1$

Step 5

Divide each term in $3 x = \frac{π}{2} + 1$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = \frac{π}{2} + 1$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{1}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{1}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{3} + \frac{1}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{3} + \frac{1}{3}$

Step 5.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{3}$ .

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Step 5.3.1.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

$x = \frac{π}{2 \cdot 3} + \frac{1}{3}$

Step 5.3.1.2.2

Multiply $2$ by $3$ .

$x = \frac{π}{6} + \frac{1}{3}$

Step 6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$3 x - 1 = 2 π - \frac{π}{2}$

Step 7

Solve for $x$ .

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Step 7.1

Simplify $2 π - \frac{π}{2}$ .

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Step 7.1.1

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

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

Step 7.1.2

Combine fractions.

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Step 7.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 7.1.2.2

Combine the numerators over the common denominator.

$3 x - 1 = \frac{2 π \cdot 2 - π}{2}$

Step 7.1.3

Simplify the numerator.

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Step 7.1.3.1

Multiply $2$ by $2$ .

$3 x - 1 = \frac{4 π - π}{2}$

Step 7.1.3.2

Subtract $π$ from $4 π$ .

$3 x - 1 = \frac{3 π}{2}$

Step 7.2

Add $1$ to both sides of the equation.

$3 x = \frac{3 π}{2} + 1$

Step 7.3

Divide each term in $3 x = \frac{3 π}{2} + 1$ by $3$ and simplify.

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Step 7.3.1

Divide each term in $3 x = \frac{3 π}{2} + 1$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{3 π}{2}}{3} + \frac{1}{3}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $3$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{3 π}{2}}{3} + \frac{1}{3}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{3} + \frac{1}{3}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Simplify each term.

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Step 7.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{3} + \frac{1}{3}$

Step 7.3.3.1.2

Cancel the common factor of $3$ .

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Step 7.3.3.1.2.1

Factor $3$ out of $3 π$ .

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

Step 7.3.3.1.2.2

Cancel the common factor.

$x = \frac{3 π}{2} \cdot \frac{1}{3} + \frac{1}{3}$

Step 7.3.3.1.2.3

Rewrite the expression.

$x = \frac{π}{2} + \frac{1}{3}$

Step 8

Find the period of $\cos (3 x - 1)$ .

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 9

The period of the $\cos (3 x - 1)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = \frac{π}{6} + \frac{1}{3} + \frac{2 π n}{3}, \frac{π}{2} + \frac{1}{3} + \frac{2 π n}{3}$ , for any integer $n$

Step 10

Consolidate the answers.

$x = \frac{π}{6} + \frac{1}{3} + \frac{π n}{3}$ , for any integer $n$