Trigonometry Examples Solving Trigonometric Equations

Solve for x

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

Step 1

Add $1$ to both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

Step 3

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 4

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

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Step 4.1

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

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

Step 4.2

Any root of $1$ is $1$ .

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

Step 4.3

Simplify the denominator.

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Step 4.3.1

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

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

Step 4.3.2

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

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

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.

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

Step 5.2

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

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

Step 5.3

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 6

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

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the right side.

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Step 7.2.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 7.3

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 7.4

Simplify $2 π - \frac{π}{3}$ .

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Step 7.4.1

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 7.4.2

Combine fractions.

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Step 7.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 7.4.3

Simplify the numerator.

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Step 7.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 7.4.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 7.5

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

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Step 7.5.1

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

$\frac{2 π}{| b |}$

Step 7.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.5.3

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

$\frac{2 π}{1}$

Step 7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.6

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 $n$

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 $\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 8.3

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 8.4

Simplify $2 π - \frac{2 π}{3}$ .

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Step 8.4.1

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 8.4.2

Combine fractions.

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Step 8.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 8.4.3

Simplify the numerator.

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Step 8.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - 2 π}{3}$

Step 8.4.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{4 π}{3}$

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 $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 8.5.2

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

$\frac{2 π}{| 1 |}$

Step 8.5.3

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

$\frac{2 π}{1}$

Step 8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 8.6

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 $n$

Step 9

List all of the solutions.

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

Step 10

Consolidate the solutions.

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Step 10.1

Consolidate $\frac{π}{3} + 2 π n$ and $\frac{4 π}{3} + 2 π n$ to $\frac{π}{3} + π n$ .

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

Step 10.2

Consolidate $\frac{5 π}{3} + 2 π n$ and $\frac{2 π}{3} + 2 π n$ to $\frac{2 π}{3} + π n$ .

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

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