Trigonometry Examples

$\cos^{2} (x) = \frac{4}{25}$

Step 1

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

$\cos (x) = \pm \sqrt{\frac{4}{25}}$

Step 2

Simplify $\pm \sqrt{\frac{4}{25}}$ .

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Step 2.1

Rewrite $\sqrt{\frac{4}{25}}$ as $\frac{\sqrt{4}}{\sqrt{25}}$ .

$\cos (x) = \pm \frac{\sqrt{4}}{\sqrt{25}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

$\cos (x) = \pm \frac{\sqrt{2^{2}}}{\sqrt{25}}$

Step 2.2.2

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

$\cos (x) = \pm \frac{2}{\sqrt{25}}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Rewrite $25$ as $5^{2}$ .

$\cos (x) = \pm \frac{2}{\sqrt{5^{2}}}$

Step 2.3.2

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

$\cos (x) = \pm \frac{2}{5}$

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.

$\cos (x) = \frac{2}{5}$

Step 3.2

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

$\cos (x) = - \frac{2}{5}$

Step 3.3

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

$\cos (x) = \frac{2}{5}, - \frac{2}{5}$

Step 4

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

$\cos (x) = \frac{2}{5}$

$\cos (x) = - \frac{2}{5}$

Step 5

Solve for $x$ in $\cos (x) = \frac{2}{5}$ .

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Step 5.1

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

$x = \arccos (\frac{2}{5})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Evaluate $\arccos (\frac{2}{5})$ .

$x = 1.15927948$

Step 5.3

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.

$x = 2 (3.14159265) - 1.15927948$

Step 5.4

Solve for $x$ .

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Step 5.4.1

Remove parentheses.

$x = 2 (3.14159265) - 1.15927948$

Step 5.4.2

Simplify $2 (3.14159265) - 1.15927948$ .

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Step 5.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.15927948$

Step 5.4.2.2

Subtract $1.15927948$ from $6.2831853$ .

$x = 5.12390582$

Step 5.5

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

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Step 5.5.1

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

$\frac{2 π}{| b |}$

Step 5.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.5.3

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

$\frac{2 π}{1}$

Step 5.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.6

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

$x = 1.15927948 + 2 π n, 5.12390582 + 2 π n$ , for any integer $n$

Step 6

Solve for $x$ in $\cos (x) = - \frac{2}{5}$ .

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Step 6.1

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

$x = \arccos (- \frac{2}{5})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Evaluate $\arccos (- \frac{2}{5})$ .

$x = 1.98231317$

Step 6.3

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.

$x = 2 (3.14159265) - 1.98231317$

Step 6.4

Solve for $x$ .

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Step 6.4.1

Remove parentheses.

$x = 2 (3.14159265) - 1.98231317$

Step 6.4.2

Simplify $2 (3.14159265) - 1.98231317$ .

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Step 6.4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.98231317$

Step 6.4.2.2

Subtract $1.98231317$ from $6.2831853$ .

$x = 4.30087213$

Step 6.5

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

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Step 6.5.1

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

$\frac{2 π}{| b |}$

Step 6.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.5.3

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

$\frac{2 π}{1}$

Step 6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.6

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

$x = 1.98231317 + 2 π n, 4.30087213 + 2 π n$ , for any integer $n$

Step 7

List all of the solutions.

$x = 1.15927948 + 2 π n, 5.12390582 + 2 π n, 1.98231317 + 2 π n, 4.30087213 + 2 π n$ , for any integer $n$

Step 8

Consolidate the solutions.

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Step 8.1

Consolidate $1.15927948 + 2 π n$ and $4.30087213 + 2 π n$ to $1.15927948 + π n$ .

$x = 1.15927948 + π n, 5.12390582 + 2 π n, 1.98231317 + 2 π n$ , for any integer $n$

Step 8.2

Consolidate $5.12390582 + 2 π n$ and $1.98231317 + 2 π n$ to $1.98231317 + π n$ .

$x = 1.15927948 + π n, 1.98231317 + π n$ , for any integer $n$