Trigonometry Examples

$9 \cos^{2} (θ) - 4 = 0$

Step 1

Add $4$ to both sides of the equation.

$9 \cos^{2} (θ) = 4$

Step 2

Divide each term in $9 \cos^{2} (θ) = 4$ by $9$ and simplify.

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Step 2.1

Divide each term in $9 \cos^{2} (θ) = 4$ by $9$ .

$\frac{9 \cos^{2} (θ)}{9} = \frac{4}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $9$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{9 \cos^{2} (θ)}{9} = \frac{4}{9}$

Step 2.2.1.2

Divide $\cos^{2} (θ)$ by $1$ .

$\cos^{2} (θ) = \frac{4}{9}$

Step 3

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

$\cos (θ) = \pm \sqrt{\frac{4}{9}}$

Step 4

Simplify $\pm \sqrt{\frac{4}{9}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$\cos (θ) = \pm \frac{\sqrt{4}}{\sqrt{9}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

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

$\cos (θ) = \pm \frac{\sqrt{2^{2}}}{\sqrt{9}}$

Step 4.2.2

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

$\cos (θ) = \pm \frac{2}{\sqrt{9}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $9$ as $3^{2}$ .

$\cos (θ) = \pm \frac{2}{\sqrt{3^{2}}}$

Step 4.3.2

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

$\cos (θ) = \pm \frac{2}{3}$

Step 5

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

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Step 5.1

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

$\cos (θ) = \frac{2}{3}$

Step 5.2

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

$\cos (θ) = - \frac{2}{3}$

Step 5.3

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

$\cos (θ) = \frac{2}{3}, - \frac{2}{3}$

Step 6

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

$\cos (θ) = \frac{2}{3}$

$\cos (θ) = - \frac{2}{3}$

Step 7

Solve for $θ$ in $\cos (θ) = \frac{2}{3}$ .

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Step 7.1

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

$θ = \arccos (\frac{2}{3})$

Step 7.2

Simplify the right side.

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Step 7.2.1

Evaluate $\arccos (\frac{2}{3})$ .

$θ = 48.1896851$

Step 7.3

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.

$θ = 360 - 48.1896851$

Step 7.4

Subtract $48.1896851$ from $360$ .

$θ = 311.81031489$

Step 7.5

Find the period of $\cos (θ)$ .

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Step 7.5.1

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

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

Step 7.5.2

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

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

Step 7.5.3

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

$\frac{360}{1}$

Step 7.5.4

Divide $360$ by $1$ .

$360$

Step 7.6

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

$θ = 48.1896851 + 360 n, 311.81031489 + 360 n$ , for any integer $n$

Step 8

Solve for $θ$ in $\cos (θ) = - \frac{2}{3}$ .

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Step 8.1

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

$θ = \arccos (- \frac{2}{3})$

Step 8.2

Simplify the right side.

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Step 8.2.1

Evaluate $\arccos (- \frac{2}{3})$ .

$θ = 131.81031489$

Step 8.3

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.

$θ = 360 - 131.81031489$

Step 8.4

Subtract $131.81031489$ from $360$ .

$θ = 228.1896851$

Step 8.5

Find the period of $\cos (θ)$ .

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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 $\cos (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 131.81031489 + 360 n, 228.1896851 + 360 n$ , for any integer $n$

Step 9

List all of the solutions.

$θ = 48.1896851 + 360 n, 311.81031489 + 360 n, 131.81031489 + 360 n, 228.1896851 + 360 n$ , for any integer $n$

Step 10

Consolidate the solutions.

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Step 10.1

Consolidate $48.1896851 + 360 n$ and $228.1896851 + 360 n$ to $48.1896851 + 180 n$ .

$θ = 48.1896851 + 180 n, 311.81031489 + 360 n, 131.81031489 + 360 n$ , for any integer $n$

Step 10.2

Consolidate $311.81031489 + 360 n$ and $131.81031489 + 360 n$ to $131.81031489 + 180 n$ .

$θ = 48.1896851 + 180 n, 131.81031489 + 180 n$ , for any integer $n$