Trigonometry Examples

cos (2 x) = - 1

$\cos (2 x) = - 1$

Step 1

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

x

$x$ from inside the cosine.

2 x = arccos (- 1)

$2 x = \arccos (- 1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arccos (- 1)

$\arccos (- 1)$ is

π

$π$ .

2 x = π

$2 x = π$

2 x = π

$2 x = π$

Step 3

Divide each term in

2 x = π

$2 x = π$ by

2

$2$ and simplify.

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Step 3.1

Divide each term in

2 x = π

$2 x = π$ by

2

$2$ .

\frac{2 x}{2} = \frac{π}{2}

$\frac{2 x}{2} = \frac{π}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

2

$2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{π}{2}$

Step 4

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

$2 π$ to find the solution in the third quadrant.

$2 x = 2 π - π$

Step 5

Solve for

$x$ .

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Step 5.1

Subtract

$π$ from

$2 π$ .

$2 x = π$

Step 5.2

Divide each term in

$2 x = π$ by

$2$ and simplify.

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Step 5.2.1

Divide each term in

$2 x = π$ by

$2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

$2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 5.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{π}{2}$

Step 6

Find the period of

$\cos (2 x)$ .

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Step 6.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 6.2

Replace

$b$ with

$2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 6.3

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

$0$ and

$2$ is

$2$ .

$\frac{2 π}{2}$

Step 6.4

Cancel the common factor of

$2$ .

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Step 6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 6.4.2

Divide

$π$ by

$1$ .

$π$

Step 7

The period of the

$\cos (2 x)$ function is

$π$ so values will repeat every

$π$ radians in both directions.

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

$n$