Trigonometry Examples

cos (θ) = 0.5

$\cos (θ) = 0.5$

Step 1

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

θ

$θ$ from inside the cosine.

θ = arccos (0.5)

$θ = \arccos (0.5)$

Step 2

Simplify the right side.

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Step 2.1

Evaluate

arccos (0.5)

$\arccos (0.5)$ .

θ = 60

$θ = 60$

θ = 60

$θ = 60$

Step 3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from

360

$360$ to find the solution in the fourth quadrant.

θ = 360 - 60

$θ = 360 - 60$

Step 4

Subtract

60

$60$ from

360

$360$ .

θ = 300

$θ = 300$

Step 5

Find the period of

cos (θ)

$\cos (θ)$ .

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Step 5.1

The period of the function can be calculated using

\frac{360}{| b |}

$\frac{360}{| b |}$ .

\frac{360}{| b |}

$\frac{360}{| b |}$

Step 5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{360}{| 1 |}

$\frac{360}{| 1 |}$

Step 5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{360}{1}

$\frac{360}{1}$

Step 5.4

Divide

360

$360$ by

1

$1$ .

360

$360$

360

$360$

Step 6

The period of the

cos (θ)

$\cos (θ)$ function is

360

$360$ so values will repeat every

360

$360$ degrees in both directions.

θ = 60 + 360 n, 300 + 360 n

$θ = 60 + 360 n, 300 + 360 n$ , for any integer

n

$n$