Algebra Examples

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

Step 1

Take the inverse cosine of both sides of the equation to extract

$x$ from inside the cosine.

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

Step 2

Simplify the right side.

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Step 2.1

The exact value of

$\arccos (\frac{\sqrt{2}}{2})$ is

$\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 3

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.

$x = 2 π - \frac{π}{4}$

Step 4

Simplify

$2 π - \frac{π}{4}$ .

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Step 4.1

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 4.2

Combine fractions.

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Step 4.2.1

Combine

$2 π$ and

$\frac{4}{4}$ .

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

Step 4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Multiply

$4$ by

$2$ .

$x = \frac{8 π - π}{4}$

Step 4.3.2

Subtract

$π$ from

$8 π$ .

$x = \frac{7 π}{4}$

Step 5

Find the period of

$\cos (x)$ .

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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.

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer

$n$