Precalculus Examples

$\cos (2 θ - \frac{π}{2}) = 1$

Step 1

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

$2 θ - \frac{π}{2} = \arccos (1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arccos (1)$ is $0$ .

$2 θ - \frac{π}{2} = 0$

Step 3

Add $\frac{π}{2}$ to both sides of the equation.

$2 θ = \frac{π}{2}$

Step 4

Divide each term in $2 θ = \frac{π}{2}$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 θ = \frac{π}{2}$ by $2$ .

$\frac{2 θ}{2} = \frac{\frac{π}{2}}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{π}{2}}{2}$

Step 4.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{2}}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{π}{2} \cdot \frac{1}{2}$

Step 4.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 4.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$θ = \frac{π}{2 \cdot 2}$

Step 4.3.2.2

Multiply $2$ by $2$ .

$θ = \frac{π}{4}$

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.

$2 θ - \frac{π}{2} = 2 π - 0$

Step 6

Solve for $θ$ .

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Step 6.1

Subtract $0$ from $2 π$ .

$2 θ - \frac{π}{2} = 2 π$

Step 6.2

Move all terms not containing $θ$ to the right side of the equation.

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Step 6.2.1

Add $\frac{π}{2}$ to both sides of the equation.

$2 θ = 2 π + \frac{π}{2}$

Step 6.2.2

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

$2 θ = 2 π \cdot \frac{2}{2} + \frac{π}{2}$

Step 6.2.3

Combine $2 π$ and $\frac{2}{2}$ .

$2 θ = \frac{2 π \cdot 2}{2} + \frac{π}{2}$

Step 6.2.4

Combine the numerators over the common denominator.

$2 θ = \frac{2 π \cdot 2 + π}{2}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $2$ by $2$ .

$2 θ = \frac{4 π + π}{2}$

Step 6.2.5.2

Add $4 π$ and $π$ .

$2 θ = \frac{5 π}{2}$

Step 6.3

Divide each term in $2 θ = \frac{5 π}{2}$ by $2$ and simplify.

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Step 6.3.1

Divide each term in $2 θ = \frac{5 π}{2}$ by $2$ .

$\frac{2 θ}{2} = \frac{\frac{5 π}{2}}{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{5 π}{2}}{2}$

Step 6.3.2.1.2

Divide $θ$ by $1$ .

$θ = \frac{\frac{5 π}{2}}{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{5 π}{2} \cdot \frac{1}{2}$

Step 6.3.3.2

Multiply $\frac{5 π}{2} \cdot \frac{1}{2}$ .

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Step 6.3.3.2.1

Multiply $\frac{5 π}{2}$ by $\frac{1}{2}$ .

$θ = \frac{5 π}{2 \cdot 2}$

Step 6.3.3.2.2

Multiply $2$ by $2$ .

$θ = \frac{5 π}{4}$

Step 7

Find the period of $\cos (2 θ - \frac{π}{2})$ .

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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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 7.4.2

Divide $π$ by $1$ .

$π$

Step 8

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

$θ = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 9

Consolidate the answers.

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