Trigonometry Examples

$\cos (4 x) = - 1$

Step 1

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

$4 x = \arccos (- 1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arccos (- 1)$ is $π$ .

$4 x = π$

Step 3

Divide each term in $4 x = π$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 x = π$ by $4$ .

$\frac{4 x}{4} = \frac{π}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{π}{4}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{4}$

Step 4

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.

$4 x = 2 π - π$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $π$ from $2 π$ .

$4 x = π$

Step 5.2

Divide each term in $4 x = π$ by $4$ and simplify.

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Step 5.2.1

Divide each term in $4 x = π$ by $4$ .

$\frac{4 x}{4} = \frac{π}{4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{π}{4}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{4}$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 6.4

Cancel the common factor of $2$ and $4$ .

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Step 6.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 6.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 6.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 7

The period of the $\cos (4 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

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