Trigonometry Examples

cos (2 x) + 2 {cos}^{2} (x) = 0

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

Step 1

Simplify the left side of the equation.

Tap for more steps...

Step 1.1

Use the double-angle identity to transform

cos (2 x)

$\cos (2 x)$ to

2 {cos}^{2} (x) - 1

$2 \cos^{2} (x) - 1$ .

2 {cos}^{2} (x) - 1 + 2 {cos}^{2} (x) = 0

$2 \cos^{2} (x) - 1 + 2 \cos^{2} (x) = 0$

Step 1.2

Add

2 {cos}^{2} (x)

$2 \cos^{2} (x)$ and

2 {cos}^{2} (x)

$2 \cos^{2} (x)$ .

- 1 + 4 {cos}^{2} (x) = 0

$- 1 + 4 \cos^{2} (x) = 0$

- 1 + 4 {cos}^{2} (x) = 0

$- 1 + 4 \cos^{2} (x) = 0$

Step 2

Factor

- 1 + 4 {cos}^{2} (x)

$- 1 + 4 \cos^{2} (x)$ .

Tap for more steps...

Step 2.1

Rewrite

4 {cos}^{2} (x)

$4 \cos^{2} (x)$ as

{(2 cos (x))}^{2}

${(2 \cos (x))}^{2}$ .

- 1 + {(2 cos (x))}^{2} = 0

$- 1 + {(2 \cos (x))}^{2} = 0$

Step 2.2

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

- 1^{2} + {(2 cos (x))}^{2} = 0

$- 1^{2} + {(2 \cos (x))}^{2} = 0$

Step 2.3

Reorder

- 1^{2}

$- 1^{2}$ and

{(2 cos (x))}^{2}

${(2 \cos (x))}^{2}$ .

{(2 cos (x))}^{2} - 1^{2} = 0

${(2 \cos (x))}^{2} - 1^{2} = 0$

Step 2.4

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 2 cos (x)

$a = 2 \cos (x)$ and

b = 1

$b = 1$ .

(2 cos (x) + 1) (2 cos (x) - 1) = 0

$(2 \cos (x) + 1) (2 \cos (x) - 1) = 0$

(2 cos (x) + 1) (2 cos (x) - 1) = 0

$(2 \cos (x) + 1) (2 \cos (x) - 1) = 0$

Step 3

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

2 cos (x) + 1 = 0

$2 \cos (x) + 1 = 0$

2 cos (x) - 1 = 0

$2 \cos (x) - 1 = 0$

Step 4

Set

2 cos (x) + 1

$2 \cos (x) + 1$ equal to

0

$0$ and solve for

x

$x$ .

Tap for more steps...

Step 4.1

Set

2 cos (x) + 1

$2 \cos (x) + 1$ equal to

0

$0$ .

2 cos (x) + 1 = 0

$2 \cos (x) + 1 = 0$

Step 4.2

Solve

2 cos (x) + 1 = 0

$2 \cos (x) + 1 = 0$ for

x

$x$ .

Tap for more steps...

Step 4.2.1

Subtract

1

$1$ from both sides of the equation.

2 cos (x) = - 1

$2 \cos (x) = - 1$

Step 4.2.2

Divide each term in

2 cos (x) = - 1

$2 \cos (x) = - 1$ by

2

$2$ and simplify.

Tap for more steps...

Step 4.2.2.1

Divide each term in

2 cos (x) = - 1

$2 \cos (x) = - 1$ by

2

$2$ .

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

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

Step 4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.1

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 4.2.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.2.1.2

Divide

$\cos (x)$ by

$1$ .

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

Step 4.2.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.2.3.1

Move the negative in front of the fraction.

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

Step 4.2.3

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

$x$ from inside the cosine.

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

Step 4.2.4

Simplify the right side.

Tap for more steps...

Step 4.2.4.1

The exact value of

$\arccos (- \frac{1}{2})$ is

$\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 4.2.5

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.

$x = 2 π - \frac{2 π}{3}$

Step 4.2.6

Simplify

$2 π - \frac{2 π}{3}$ .

Tap for more steps...

Step 4.2.6.1

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 4.2.6.2

Combine fractions.

Tap for more steps...

Step 4.2.6.2.1

Combine

$2 π$ and

$\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Step 4.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 4.2.6.3

Simplify the numerator.

Tap for more steps...

Step 4.2.6.3.1

Multiply

$3$ by

$2$ .

$x = \frac{6 π - 2 π}{3}$

Step 4.2.6.3.2

Subtract

$2 π$ from

$6 π$ .

$x = \frac{4 π}{3}$

Step 4.2.7

Find the period of

$\cos (x)$ .

Tap for more steps...

Step 4.2.7.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.2.7.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 4.2.7.3

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

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 4.2.7.4

Divide

$2 π$ by

$1$ .

$2 π$

Step 4.2.8

The period of the

$\cos (x)$ function is

$2 π$ so values will repeat every

$2 π$ radians in both directions.

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer

$n$

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer

$n$

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer

$n$

Step 5

Set

$2 \cos (x) - 1$ equal to

$0$ and solve for

$x$ .

Tap for more steps...

Step 5.1

Set

$2 \cos (x) - 1$ equal to

$0$ .

$2 \cos (x) - 1 = 0$

Step 5.2

Solve

$2 \cos (x) - 1 = 0$ for

$x$ .

Tap for more steps...

Step 5.2.1

Add

$1$ to both sides of the equation.

$2 \cos (x) = 1$

Step 5.2.2

Divide each term in

$2 \cos (x) = 1$ by

$2$ and simplify.

Tap for more steps...

Step 5.2.2.1

Divide each term in

$2 \cos (x) = 1$ by

$2$ .

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

Step 5.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.2.1

Cancel the common factor of

$2$ .

Tap for more steps...

Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide

$\cos (x)$ by

$1$ .

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

Step 5.2.3

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

$x$ from inside the cosine.

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

Step 5.2.4

Simplify the right side.

Tap for more steps...

Step 5.2.4.1

The exact value of

$\arccos (\frac{1}{2})$ is

$\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 5.2.5

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.

$x = 2 π - \frac{π}{3}$

Step 5.2.6

Simplify

$2 π - \frac{π}{3}$ .

Tap for more steps...

Step 5.2.6.1

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 5.2.6.2

Combine fractions.

Tap for more steps...

Step 5.2.6.2.1

Combine

$2 π$ and

$\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 5.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 5.2.6.3

Simplify the numerator.

Tap for more steps...

Step 5.2.6.3.1

Multiply

$3$ by

$2$ .

$x = \frac{6 π - π}{3}$

Step 5.2.6.3.2

Subtract

$π$ from

$6 π$ .

$x = \frac{5 π}{3}$

Step 5.2.7

Find the period of

$\cos (x)$ .

Tap for more steps...

Step 5.2.7.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.2.7.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.2.7.3

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

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 5.2.7.4

Divide

$2 π$ by

$1$ .

$2 π$

Step 5.2.8

The period of the

$\cos (x)$ function is

$2 π$ so values will repeat every

$2 π$ radians in both directions.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer

$n$

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer

$n$

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer

$n$

Step 6

The final solution is all the values that make

$(2 \cos (x) + 1) (2 \cos (x) - 1) = 0$ true.

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n, \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer

$n$

Step 7

Consolidate the answers.

Tap for more steps...

Step 7.1

Consolidate

$\frac{2 π}{3} + 2 π n$ and

$\frac{5 π}{3} + 2 π n$ to

$\frac{2 π}{3} + π n$ .

$x = \frac{2 π}{3} + π n, \frac{4 π}{3} + 2 π n, \frac{π}{3} + 2 π n$ , for any integer

$n$

Step 7.2

Consolidate

$\frac{4 π}{3} + 2 π n$ and

$\frac{π}{3} + 2 π n$ to

$\frac{π}{3} + π n$ .

$x = \frac{2 π}{3} + π n, \frac{π}{3} + π n$ , for any integer

$n$

$x = \frac{2 π}{3} + π n, \frac{π}{3} + π n$ , for any integer

$n$