Precalculus Examples

$\cos (3 x) = 1$

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

Add $1$ to both sides of the equation.

$\cos (3 x) = 1$

Step 3

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

$3 x = \arccos (1)$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (1)$ is $0$ .

$3 x = 0$

Step 5

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

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.

$\frac{3 x}{3} = \frac{0}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $3$ .

$x = 0$

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.

$3 x = 2 π - 0$

Step 7

Solve for $x$ .

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Step 7.1

Simplify.

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Step 7.1.1

Multiply $- 1$ by $0$ .

$3 x = 2 π + 0$

Step 7.1.2

Add $2 π$ and $0$ .

$3 x = 2 π$

Step 7.2

Divide each term in $3 x = 2 π$ by $3$ and simplify.

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Step 7.2.1

Divide each term in $3 x = 2 π$ by $3$ .

$\frac{3 x}{3} = \frac{2 π}{3}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2 π}{3}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{3}$

Step 8

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

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

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

Step 10

Consolidate the answers.

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