Trigonometry Examples

cos (\frac{θ}{2}) = - \frac{\sqrt{2}}{2}

$\cos (\frac{θ}{2}) = - \frac{\sqrt{2}}{2}$

Step 1

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

θ

$θ$ from inside the cosine.

\frac{θ}{2} = arccos (- \frac{\sqrt{2}}{2})

$\frac{θ}{2} = \arccos (- \frac{\sqrt{2}}{2})$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arccos (- \frac{\sqrt{2}}{2})

$\arccos (- \frac{\sqrt{2}}{2})$ is

135

$135$ .

\frac{θ}{2} = 135

$\frac{θ}{2} = 135$

\frac{θ}{2} = 135

$\frac{θ}{2} = 135$

Step 3

Multiply both sides of the equation by

2

$2$ .

2 \frac{θ}{2} = 2 \cdot 135

$2 \frac{θ}{2} = 2 \cdot 135$

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

2

$2$ .

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Step 4.1.1.1

Cancel the common factor.

$2 \frac{θ}{2} = 2 \cdot 135$

Step 4.1.1.2

Rewrite the expression.

$θ = 2 \cdot 135$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply

$2$ by

$135$ .

$θ = 270$

Step 5

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from

$360$ to find the solution in the third quadrant.

$\frac{θ}{2} = 360 - 135$

Step 6

Solve for

$θ$ .

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Step 6.1

Multiply both sides of the equation by

$2$ .

$2 \frac{θ}{2} = 2 (360 - 135)$

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

$2$ .

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Step 6.2.1.1.1

Cancel the common factor.

$2 \frac{θ}{2} = 2 (360 - 135)$

Step 6.2.1.1.2

Rewrite the expression.

$θ = 2 (360 - 135)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify

$2 (360 - 135)$ .

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Step 6.2.2.1.1

Subtract

$135$ from

$360$ .

$θ = 2 \cdot 225$

Step 6.2.2.1.2

Multiply

$2$ by

$225$ .

$θ = 450$

Step 7

Find the period of

$\cos (\frac{θ}{2})$ .

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Step 7.1

The period of the function can be calculated using

$\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 7.2

Replace

$b$ with

$\frac{1}{2}$ in the formula for period.

$\frac{360}{| \frac{1}{2} |}$

Step 7.3

$\frac{1}{2}$ is approximately

$0.5$ which is positive so remove the absolute value

$\frac{360}{\frac{1}{2}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$360 \cdot 2$

Step 7.5

Multiply

$360$ by

$2$ .

$720$

Step 8

The period of the

$\cos (\frac{θ}{2})$ function is

$720$ so values will repeat every

$720$ degrees in both directions.

$θ = 270 + 720 n, 450 + 720 n$ , for any integer

$n$