Precalculus Examples

\cos (z) = 0

$\cos (z) = 0$

Step 1

Take the inverse cosine of both sides of the equation to extract

z

$z$ from inside the cosine.

z = \arccos (0)

$z = \arccos (0)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

\arccos (0)

$\arccos (0)$ is

\frac{π}{2}

$\frac{π}{2}$ .

z = \frac{π}{2}

$z = \frac{π}{2}$

z = \frac{π}{2}

$z = \frac{π}{2}$

Step 3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from

2 π

$2 π$ to find the solution in the fourth quadrant.

z = 2 π - \frac{π}{2}

$z = 2 π - \frac{π}{2}$

Step 4

Simplify

2 π - \frac{π}{2}

$2 π - \frac{π}{2}$ .

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Step 4.1

To write

2 π

$2 π$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

z = 2 π \cdot \frac{2}{2} - \frac{π}{2}

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

Step 4.2

Combine fractions.

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Step 4.2.1

Combine

2 π

$2 π$ and

\frac{2}{2}

$\frac{2}{2}$ .

z = \frac{2 π \cdot 2}{2} - \frac{π}{2}

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

Step 4.2.2

Combine the numerators over the common denominator.

z = \frac{2 π \cdot 2 - π}{2}

$z = \frac{2 π \cdot 2 - π}{2}$

z = \frac{2 π \cdot 2 - π}{2}

$z = \frac{2 π \cdot 2 - π}{2}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Multiply

2

$2$ by

2

$2$ .

z = \frac{4 π - π}{2}

$z = \frac{4 π - π}{2}$

Step 4.3.2

Subtract

π

$π$ from

4 π

$4 π$ .

z = \frac{3 π}{2}

$z = \frac{3 π}{2}$

z = \frac{3 π}{2}

$z = \frac{3 π}{2}$

z = \frac{3 π}{2}

$z = \frac{3 π}{2}$

Step 5

Find the period of

\cos (z)

$\cos (z)$ .

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Step 5.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 5.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 6

The period of the

\cos (z)

$\cos (z)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

z = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n

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

n

$n$

Step 7

Consolidate the answers.

z = \frac{π}{2} + π n

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

n

$n$