Trigonometry Examples

$\cos (x) = \frac{5}{8}$

Step 1

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

$x$ from inside the cosine.

$x = \arccos (\frac{5}{8})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate

$\arccos (\frac{5}{8})$ .

$x = 0.89566479$

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.

$x = 2 (3.14159265) - 0.89566479$

Step 4

Simplify

$2 (3.14159265) - 0.89566479$ .

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Step 4.1

Multiply

$2$ by

$3.14159265$ .

$x = 6.2831853 - 0.89566479$

Step 4.2

Subtract

$0.89566479$ from

$6.2831853$ .

$x = 5.38752051$

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.

$x = 0.89566479 + 2 π n, 5.38752051 + 2 π n$ , for any integer

$n$