Trigonometry Examples

$\cos (x) + \sqrt{3} = - \cos (x)$

Step 1

Move all terms containing

$\cos (x)$ to the left side of the equation.

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Step 1.1

Add

$\cos (x)$ to both sides of the equation.

$\cos (x) + \sqrt{3} + \cos (x) = 0$

Step 1.2

Add

$\cos (x)$ and

$\cos (x)$ .

$2 \cos (x) + \sqrt{3} = 0$

Step 2

Subtract

$\sqrt{3}$ from both sides of the equation.

$2 \cos (x) = - \sqrt{3}$

Step 3

Divide each term in

$2 \cos (x) = - \sqrt{3}$ by

$2$ and simplify.

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Step 3.1

Divide each term in

$2 \cos (x) = - \sqrt{3}$ by

$2$ .

$\frac{2 \cos (x)}{2} = \frac{- \sqrt{3}}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

$2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 \cos (x)}{2} = \frac{- \sqrt{3}}{2}$

Step 3.2.1.2

Divide

$\cos (x)$ by

$1$ .

$\cos (x) = \frac{- \sqrt{3}}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$\cos (x) = - \frac{\sqrt{3}}{2}$

Step 4

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

$x$ from inside the cosine.

$x = \arccos (- \frac{\sqrt{3}}{2})$

Step 5

Simplify the right side.

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Step 5.1

The exact value of

$\arccos (- \frac{\sqrt{3}}{2})$ is

$150$ .

$x = 150$

Step 6

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.

$x = 360 - 150$

Step 7

Subtract

$150$ from

$360$ .

$x = 210$

Step 8

Find the period of

$\cos (x)$ .

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Step 8.1

The period of the function can be calculated using

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

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

Step 8.2

Replace

$b$ with

$1$ in the formula for period.

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

Step 8.3

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

$0$ and

$1$ is

$1$ .

$\frac{360}{1}$

Step 8.4

Divide

$360$ by

$1$ .

$360$

Step 9

The period of the

$\cos (x)$ function is

$360$ so values will repeat every

$360$ degrees in both directions.

$x = 150 + 360 n, 210 + 360 n$ , for any integer

$n$