Precalculus Examples

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

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $2$ .

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

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $2$ .

$\cos (x) \cdot 2$

Step 3

Move $2$ to the left of $\cos (x)$ .

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

Step 4

Simplify $\cos (x) \cdot \frac{2}{2}$ .

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Step 4.1

Divide $2$ by $2$ .

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

Step 4.2

Multiply $\cos (x)$ by $1$ .

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

Step 5

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

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

Step 6

Simplify the right side.

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Step 6.1

The exact value of $\arccos (- \frac{\sqrt{2}}{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 7

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.

$x = 2 π - \frac{3 π}{4}$

Step 8

Simplify $2 π - \frac{3 π}{4}$ .

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Step 8.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 8.2

Combine fractions.

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Step 8.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 8.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 8.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 9

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

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

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

$\frac{2 π}{| 1 |}$

Step 9.3

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

$\frac{2 π}{1}$

Step 9.4

Divide $2 π$ by $1$ .

$2 π$

Step 10

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

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$