Trigonometry Examples

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

Step 1

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

$r = \pm \sqrt{- 4 \cos (2 x)}$

Step 2

Simplify $\pm \sqrt{- 4 \cos (2 x)}$ .

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Step 2.1

Rewrite $- 4 \cos (2 x)$ as $2^{2} \cdot (- 1 \cos (2 x))$ .

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Step 2.1.1

Factor $4$ out of $- 4$ .

$r = \pm \sqrt{4 (- 1) \cos (2 x)}$

Step 2.1.2

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

$r = \pm \sqrt{2^{2} \cdot - 1 \cos (2 x)}$

Step 2.1.3

Add parentheses.

$r = \pm \sqrt{2^{2} \cdot (- 1 \cos (2 x))}$

Step 2.2

Pull terms out from under the radical.

$r = \pm 2 \sqrt{- 1 \cos (2 x)}$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

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

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

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

Step 4

Set the radicand in $\sqrt{- 1 \cos (2 x)}$ greater than or equal to $0$ to find where the expression is defined.

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

Step 5

Solve for $x$ .

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Step 5.1

Divide each term in $- 1 \cos (2 x) \geq 0$ by $- 1$ and simplify.

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Step 5.1.1

Divide each term in $- 1 \cos (2 x) \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 1 \cos (2 x)}{- 1} \leq \frac{0}{- 1}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (2 x)}{1} \leq \frac{0}{- 1}$

Step 5.1.2.2

Divide $\cos (2 x)$ by $1$ .

$\cos (2 x) \leq \frac{0}{- 1}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $- 1$ .

$\cos (2 x) \leq 0$

Step 5.2

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

$2 x \leq \arccos (0)$

Step 5.3

Simplify the right side.

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Step 5.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$2 x \leq \frac{π}{2}$

Step 5.4

Divide each term in $2 x \leq \frac{π}{2}$ by $2$ and simplify.

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Step 5.4.1

Divide each term in $2 x \leq \frac{π}{2}$ by $2$ .

$\frac{2 x}{2} \leq \frac{\frac{π}{2}}{2}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{\frac{π}{2}}{2}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{\frac{π}{2}}{2}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \leq \frac{π}{2} \cdot \frac{1}{2}$

Step 5.4.3.2

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

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Step 5.4.3.2.1

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

$x \leq \frac{π}{2 \cdot 2}$

Step 5.4.3.2.2

Multiply $2$ by $2$ .

$x \leq \frac{π}{4}$

Step 5.5

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.

$2 x = 2 π - \frac{π}{2}$

Step 5.6

Solve for $x$ .

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Step 5.6.1

Simplify.

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Step 5.6.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.6.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.6.1.3

Combine the numerators over the common denominator.

$2 x = \frac{2 π \cdot 2 - π}{2}$

Step 5.6.1.4

Multiply $2$ by $2$ .

$2 x = \frac{4 π - π}{2}$

Step 5.6.1.5

Subtract $π$ from $4 π$ .

$2 x = \frac{3 π}{2}$

Step 5.6.2

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ and simplify.

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Step 5.6.2.1

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

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

Step 5.6.2.2

Simplify the left side.

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Step 5.6.2.2.1

Cancel the common factor of $2$ .

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Step 5.6.2.2.1.1

Cancel the common factor.

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

Step 5.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 5.6.2.3

Simplify the right side.

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Step 5.6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 5.6.2.3.2

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

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Step 5.6.2.3.2.1

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

$x = \frac{3 π}{2 \cdot 2}$

Step 5.6.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 5.7

Find the period of $\cos (2 x)$ .

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Step 5.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.7.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 5.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 5.7.4.2

Divide $π$ by $1$ .

$π$

Step 5.8

The period of the $\cos (2 x)$ function is $π$ so values will repeat every $π$ radians in both directions.

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

Step 5.9

Consolidate the answers.

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

Step 5.10

Use each root to create test intervals.

$\frac{π}{4} < x < \frac{3 π}{4}$

$\frac{3 π}{4} < x < \frac{5 π}{4}$

Step 5.11

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 5.11.1

Test a value on the interval $\frac{π}{4} < x < \frac{3 π}{4}$ to see if it makes the inequality true.

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Step 5.11.1.1

Choose a value on the interval $\frac{π}{4} < x < \frac{3 π}{4}$ and see if this value makes the original inequality true.

$x = 2$

Step 5.11.1.2

Replace $x$ with $2$ in the original inequality.

$- 1 \cos (2 (2)) \geq 0$

Step 5.11.1.3

The left side $0.65364362$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 5.11.2

Test a value on the interval $\frac{3 π}{4} < x < \frac{5 π}{4}$ to see if it makes the inequality true.

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Step 5.11.2.1

Choose a value on the interval $\frac{3 π}{4} < x < \frac{5 π}{4}$ and see if this value makes the original inequality true.

$x = 3$

Step 5.11.2.2

Replace $x$ with $3$ in the original inequality.

$- 1 \cos (2 (3)) \geq 0$

Step 5.11.2.3

The left side $- 0.96017028$ is less than the right side $0$ , which means that the given statement is false.

False

Step 5.11.3

Compare the intervals to determine which ones satisfy the original inequality.

$\frac{π}{4} < x < \frac{3 π}{4}$ True

$\frac{3 π}{4} < x < \frac{5 π}{4}$ False

$\frac{π}{4} < x < \frac{3 π}{4}$ True

$\frac{3 π}{4} < x < \frac{5 π}{4}$ False

Step 5.12

The solution consists of all of the true intervals.

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

Step 6

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x = \frac{π}{4} + π n, x = \frac{3 π}{4} + π n}$

Step 7