Precalculus Examples

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

Step 1

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

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

Step 2

Simplify the right side.

Tap for more steps...

Step 2.1

Evaluate $\arccos (- \frac{2}{3})$ .

$x = 2.30052398$

Step 3

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.

$x = 2 (3.14159265) - 2.30052398$

Step 4

Simplify $2 (3.14159265) - 2.30052398$ .

Tap for more steps...

Step 4.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 2.30052398$

Step 4.2

Subtract $2.30052398$ from $6.2831853$ .

$x = 3.98266132$

Step 5

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

Tap for more steps...

Step 5.1

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

$\frac{2 π}{| b |}$

Step 5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6

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

$x = 2.30052398 + 2 π n, 3.98266132 + 2 π n$ , for any integer $n$