Trigonometry Examples

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

Step 1

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

$x = \arccos (- \frac{1}{2})$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arccos (- \frac{1}{2})$ is $120$ .

$x = 120$

Step 3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the third quadrant.

$x = 360 - 120$

Step 4

Subtract $120$ from $360$ .

$x = 240$

Step 5

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

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Step 5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

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

Step 5.2

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

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

Step 5.3

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

$\frac{360}{1}$

Step 5.4

Divide $360$ by $1$ .

$360$

Step 6

The period of the $\cos (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 120 + 360 n, 240 + 360 n$ , for any integer $n$