Trigonometry Examples

$\cos (\frac{x}{4}) = 0$

Step 1

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

$\frac{x}{4} = \arccos (0)$

Step 2

Simplify the right side.

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Step 2.1

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

$\frac{x}{4} = \frac{π}{2}$

Step 3

Multiply both sides of the equation by $4$ .

$4 \frac{x}{4} = 4 \frac{π}{2}$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $4$ .

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Step 4.1.1.1

Cancel the common factor.

$4 \frac{x}{4} = 4 \frac{π}{2}$

Step 4.1.1.2

Rewrite the expression.

$x = 4 \frac{π}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Factor $2$ out of $4$ .

$x = 2 (2) \frac{π}{2}$

Step 4.2.1.2

Cancel the common factor.

$x = 2 \cdot 2 \frac{π}{2}$

Step 4.2.1.3

Rewrite the expression.

$x = 2 π$

Step 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.

$\frac{x}{4} = 2 π - \frac{π}{2}$

Step 6

Solve for $x$ .

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Step 6.1

Multiply both sides of the equation by $4$ .

$4 \frac{x}{4} = 4 (2 π - \frac{π}{2})$

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Cancel the common factor of $4$ .

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Step 6.2.1.1.1

Cancel the common factor.

$4 \frac{x}{4} = 4 (2 π - \frac{π}{2})$

Step 6.2.1.1.2

Rewrite the expression.

$x = 4 (2 π - \frac{π}{2})$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $4 (2 π - \frac{π}{2})$ .

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Step 6.2.2.1.1

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

$x = 4 (2 π \cdot \frac{2}{2} - \frac{π}{2})$

Step 6.2.2.1.2

Combine $2 π$ and $\frac{2}{2}$ .

$x = 4 (\frac{2 π \cdot 2}{2} - \frac{π}{2})$

Step 6.2.2.1.3

Combine the numerators over the common denominator.

$x = 4 \frac{2 π \cdot 2 - π}{2}$

Step 6.2.2.1.4

Cancel the common factor of $2$ .

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Step 6.2.2.1.4.1

Factor $2$ out of $4$ .

$x = 2 (2) \frac{2 π \cdot 2 - π}{2}$

Step 6.2.2.1.4.2

Cancel the common factor.

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

Step 6.2.2.1.4.3

Rewrite the expression.

$x = 2 (2 π \cdot 2 - π)$

Step 6.2.2.1.5

Multiply $2$ by $2$ .

$x = 2 (4 π - π)$

Step 6.2.2.1.6

Subtract $π$ from $4 π$ .

$x = 2 (3 π)$

Step 6.2.2.1.7

Multiply $3$ by $2$ .

$x = 6 π$

Step 7

Find the period of $\cos (\frac{x}{4})$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

Step 7.3

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Step 7.5

Multiply $4$ by $2$ .

$8 π$

Step 8

The period of the $\cos (\frac{x}{4})$ function is $8 π$ so values will repeat every $8 π$ radians in both directions.

$x = 2 π + 8 π n, 6 π + 8 π n$ , for any integer $n$

Step 9

Consolidate the answers.

$x = 2 π + 4 π n$ , for any integer $n$