Trigonometry Examples

$2 (1 - \cos^{2} (x)) = \frac{3}{2}$

Step 1

Divide each term in $2 (1 - \cos^{2} (x)) = \frac{3}{2}$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 (1 - \cos^{2} (x)) = \frac{3}{2}$ by $2$ .

$\frac{2 (1 - \cos^{2} (x))}{2} = \frac{\frac{3}{2}}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 (1 - \cos^{2} (x))}{2} = \frac{\frac{3}{2}}{2}$

Step 1.2.1.2

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

$1 - \cos^{2} (x) = \frac{\frac{3}{2}}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply the numerator by the reciprocal of the denominator.

$1 - \cos^{2} (x) = \frac{3}{2} \cdot \frac{1}{2}$

Step 1.3.2

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Step 1.3.2.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$1 - \cos^{2} (x) = \frac{3}{2 \cdot 2}$

Step 1.3.2.2

Multiply $2$ by $2$ .

$1 - \cos^{2} (x) = \frac{3}{4}$

Step 2

Move all terms not containing $\cos (x)$ to the right side of the equation.

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Step 2.1

Subtract $1$ from both sides of the equation.

$- \cos^{2} (x) = \frac{3}{4} - 1$

Step 2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \cos^{2} (x) = \frac{3}{4} - 1 \cdot \frac{4}{4}$

Step 2.3

Combine $- 1$ and $\frac{4}{4}$ .

$- \cos^{2} (x) = \frac{3}{4} + \frac{- 1 \cdot 4}{4}$

Step 2.4

Combine the numerators over the common denominator.

$- \cos^{2} (x) = \frac{3 - 1 \cdot 4}{4}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $4$ .

$- \cos^{2} (x) = \frac{3 - 4}{4}$

Step 2.5.2

Subtract $4$ from $3$ .

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

Step 2.6

Move the negative in front of the fraction.

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

Step 3

Divide each term in $- \cos^{2} (x) = - \frac{1}{4}$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \cos^{2} (x) = - \frac{1}{4}$ by $- 1$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2

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

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Dividing two negative values results in a positive value.

$\cos^{2} (x) = \frac{\frac{1}{4}}{1}$

Step 3.3.2

Divide $\frac{1}{4}$ by $1$ .

$\cos^{2} (x) = \frac{1}{4}$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{\frac{1}{4}}$

Step 5

Simplify $\pm \sqrt{\frac{1}{4}}$ .

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Step 5.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\cos (x) = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 5.2

Any root of $1$ is $1$ .

$\cos (x) = \pm \frac{1}{\sqrt{4}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 5.3.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 7

Set up each of the solutions to solve for $x$ .

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

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

Step 8

Solve for $x$ in $\cos (x) = \frac{1}{2}$ .

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Step 8.1

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

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

Step 8.2

Simplify the right side.

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Step 8.2.1

The exact value of $\arccos (\frac{1}{2})$ is $60$ .

$x = 60$

Step 8.3

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

$x = 360 - 60$

Step 8.4

Subtract $60$ from $360$ .

$x = 300$

Step 8.5

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

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Step 8.5.1

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

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

Step 8.5.2

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

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

Step 8.5.3

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

$\frac{360}{1}$

Step 8.5.4

Divide $360$ by $1$ .

$360$

Step 8.6

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

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

Step 9

Solve for $x$ in $\cos (x) = - \frac{1}{2}$ .

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Step 9.1

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

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

Step 9.2

Simplify the right side.

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Step 9.2.1

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

$x = 120$

Step 9.3

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 9.4

Subtract $120$ from $360$ .

$x = 240$

Step 9.5

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

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Step 9.5.1

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

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

Step 9.5.2

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

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

Step 9.5.3

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

$\frac{360}{1}$

Step 9.5.4

Divide $360$ by $1$ .

$360$

Step 9.6

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 $n$

Step 10

List all of the solutions.

$x = 60 + 360 n, 300 + 360 n, 120 + 360 n, 240 + 360 n$ , for any integer $n$

Step 11

Consolidate the solutions.

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Step 11.1

Consolidate $60 + 360 n$ and $240 + 360 n$ to $60 + 180 n$ .

$x = 60 + 180 n, 300 + 360 n, 120 + 360 n$ , for any integer $n$

Step 11.2

Consolidate $300 + 360 n$ and $120 + 360 n$ to $120 + 180 n$ .

$x = 60 + 180 n, 120 + 180 n$ , for any integer $n$