Precalculus Examples

$7 \cos (6 x) = 3$

Step 1

Divide each term in $7 \cos (6 x) = 3$ by $7$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $7 \cos (6 x) = 3$ by $7$ .

$\frac{7 \cos (6 x)}{7} = \frac{3}{7}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{7 \cos (6 x)}{7} = \frac{3}{7}$

Step 1.2.1.2

Divide $\cos (6 x)$ by $1$ .

$\cos (6 x) = \frac{3}{7}$

Step 2

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

$6 x = \arccos (\frac{3}{7})$

Step 3

Simplify the right side.

Tap for more steps...

Step 3.1

Evaluate $\arccos (\frac{3}{7})$ .

$6 x = 1.12788528$

Step 4

Divide each term in $6 x = 1.12788528$ by $6$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $6 x = 1.12788528$ by $6$ .

$\frac{6 x}{6} = \frac{1.12788528}{6}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{1.12788528}{6}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.12788528}{6}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Divide $1.12788528$ by $6$ .

$x = 0.18798088$

Step 5

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.

$6 x = 2 (3.14159265) - 1.12788528$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Simplify.

Tap for more steps...

Step 6.1.1

Multiply $2$ by $3.14159265$ .

$6 x = 6.2831853 - 1.12788528$

Step 6.1.2

Subtract $1.12788528$ from $6.2831853$ .

$6 x = 5.15530002$

Step 6.2

Divide each term in $6 x = 5.15530002$ by $6$ and simplify.

Tap for more steps...

Step 6.2.1

Divide each term in $6 x = 5.15530002$ by $6$ .

$\frac{6 x}{6} = \frac{5.15530002}{6}$

Step 6.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 6.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{5.15530002}{6}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5.15530002}{6}$

Step 6.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.3.1

Divide $5.15530002$ by $6$ .

$x = 0.85921667$

Step 7

Find the period of $\cos (6 x)$ .

Tap for more steps...

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 $6$ in the formula for period.

$\frac{2 π}{| 6 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{2 π}{6}$

Step 7.4

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 7.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Step 7.4.2

Cancel the common factors.

Tap for more steps...

Step 7.4.2.1

Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Step 7.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Step 7.4.2.3

Rewrite the expression.

$\frac{π}{3}$

Step 8

The period of the $\cos (6 x)$ function is $\frac{π}{3}$ so values will repeat every $\frac{π}{3}$ radians in both directions.

$x = 0.18798088 + \frac{π n}{3}, 0.85921667 + \frac{π n}{3}$ , for any integer $n$