Trigonometry Examples

cos (2 θ - \frac{π}{2}) = - 1

$\cos (2 θ - \frac{π}{2}) = - 1$

Step 1

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

θ

$θ$ from inside the cosine.

2 θ - \frac{π}{2} = arccos (- 1)

$2 θ - \frac{π}{2} = \arccos (- 1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arccos (- 1)

$\arccos (- 1)$ is

π

$π$ .

2 θ - \frac{π}{2} = π

$2 θ - \frac{π}{2} = π$

2 θ - \frac{π}{2} = π

$2 θ - \frac{π}{2} = π$

Step 3

Move all terms not containing

θ

$θ$ to the right side of the equation.

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Step 3.1

Add

\frac{π}{2}

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

2 θ = π + \frac{π}{2}

$2 θ = π + \frac{π}{2}$

Step 3.2

To write

π

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

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 3.3

Combine

π

$π$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 3.4

Combine the numerators over the common denominator.

2 θ = \frac{π \cdot 2 + π}{2}

$2 θ = \frac{π \cdot 2 + π}{2}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Move

2

$2$ to the left of

π

$π$ .

2 θ = \frac{2 \cdot π + π}{2}

$2 θ = \frac{2 \cdot π + π}{2}$

Step 3.5.2

Add

2 π

$2 π$ and

π

$π$ .

2 θ = \frac{3 π}{2}

$2 θ = \frac{3 π}{2}$

2 θ = \frac{3 π}{2}

$2 θ = \frac{3 π}{2}$

2 θ = \frac{3 π}{2}

$2 θ = \frac{3 π}{2}$

Step 4

Divide each term in

2 θ = \frac{3 π}{2}

$2 θ = \frac{3 π}{2}$ by

2

$2$ and simplify.

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Step 4.1

Divide each term in

2 θ = \frac{3 π}{2}

$2 θ = \frac{3 π}{2}$ by

2

$2$ .

\frac{2 θ}{2} = \frac{\frac{3 π}{2}}{2}

$\frac{2 θ}{2} = \frac{\frac{3 π}{2}}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

2

$2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{3 π}{2}}{2}$

Step 4.2.1.2

Divide

$θ$ by

$1$ .

$θ = \frac{\frac{3 π}{2}}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 4.3.2

Multiply

$\frac{3 π}{2} \cdot \frac{1}{2}$ .

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Step 4.3.2.1

Multiply

$\frac{3 π}{2}$ by

$\frac{1}{2}$ .

$θ = \frac{3 π}{2 \cdot 2}$

Step 4.3.2.2

Multiply

$2$ by

$2$ .

$θ = \frac{3 π}{4}$

Step 5

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.

$2 θ - \frac{π}{2} = 2 π - π$

Step 6

Solve for

$θ$ .

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Step 6.1

Subtract

$π$ from

$2 π$ .

$2 θ - \frac{π}{2} = π$

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.

$2 θ = π + \frac{π}{2}$

Step 6.2.2

To write

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

$\frac{2}{2}$ .

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

Step 6.2.3

Combine

$π$ and

$\frac{2}{2}$ .

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

Step 6.2.4

Combine the numerators over the common denominator.

$2 θ = \frac{π \cdot 2 + π}{2}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Move

$2$ to the left of

$π$ .

$2 θ = \frac{2 \cdot π + π}{2}$

Step 6.2.5.2

Add

$2 π$ and

$π$ .

$2 θ = \frac{3 π}{2}$

Step 6.3

Divide each term in

$2 θ = \frac{3 π}{2}$ by

$2$ and simplify.

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Step 6.3.1

Divide each term in

$2 θ = \frac{3 π}{2}$ by

$2$ .

$\frac{2 θ}{2} = \frac{\frac{3 π}{2}}{2}$

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.

$\frac{2 θ}{2} = \frac{\frac{3 π}{2}}{2}$

Step 6.3.2.1.2

Divide

$θ$ by

$1$ .

$θ = \frac{\frac{3 π}{2}}{2}$

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.

$θ = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 6.3.3.2

Multiply

$\frac{3 π}{2} \cdot \frac{1}{2}$ .

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Step 6.3.3.2.1

Multiply

$\frac{3 π}{2}$ by

$\frac{1}{2}$ .

$θ = \frac{3 π}{2 \cdot 2}$

Step 6.3.3.2.2

Multiply

$2$ by

$2$ .

$θ = \frac{3 π}{4}$

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.

$θ = \frac{3 π}{4} + π n$ , for any integer

$n$