Trigonometry Examples

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \cos (67.5)}{2}}$

Step 1

Simplify $\sqrt{\frac{1 + \cos (67.5)}{2}}$ .

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Step 1.1

The exact value of $\cos (67.5)$ is $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

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Step 1.1.1

Rewrite $67.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \cos (\frac{135}{2})}{2}}$

Step 1.1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$\cos (\frac{x}{2}) = \sqrt{\frac{1 \pm \sqrt{\frac{1 + \cos (135)}{2}}}{2}}$

Step 1.1.3

Change the $\pm$ to $+$ because cosine is positive in the first quadrant.

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \sqrt{\frac{1 + \cos (135)}{2}}}{2}}$

Step 1.1.4

Simplify $\sqrt{\frac{1 + \cos (135)}{2}}$ .

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Step 1.1.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \sqrt{\frac{1 - \cos (45)}{2}}}{2}}$

Step 1.1.4.2

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}}{2}}$

Step 1.1.4.3

Write $1$ as a fraction with a common denominator.

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}}}{2}}$

Step 1.1.4.4

Combine the numerators over the common denominator.

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}}{2}}$

Step 1.1.4.5

Multiply the numerator by the reciprocal of the denominator.

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}}}{2}}$

Step 1.1.4.6

Multiply $\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.1.4.6.1

Multiply $\frac{2 - \sqrt{2}}{2}$ by $\frac{1}{2}$ .

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}}}{2}}$

Step 1.1.4.6.2

Multiply $2$ by $2$ .

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \sqrt{\frac{2 - \sqrt{2}}{4}}}{2}}$

Step 1.1.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}}{2}}$

Step 1.1.4.8

Simplify the denominator.

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Step 1.1.4.8.1

Rewrite $4$ as $2^{2}$ .

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}}}{2}}$

Step 1.1.4.8.2

Pull terms out from under the radical, assuming positive real numbers.

$\cos (\frac{x}{2}) = \sqrt{\frac{1 + \frac{\sqrt{2 - \sqrt{2}}}{2}}{2}}$

Step 1.2

Write $1$ as a fraction with a common denominator.

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$\cos (\frac{x}{2}) = \sqrt{\frac{2 + \sqrt{2 - \sqrt{2}}}{2} \cdot \frac{1}{2}}$

Step 1.5

Multiply $\frac{2 + \sqrt{2 - \sqrt{2}}}{2} \cdot \frac{1}{2}$ .

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Step 1.5.1

Multiply $\frac{2 + \sqrt{2 - \sqrt{2}}}{2}$ by $\frac{1}{2}$ .

$\cos (\frac{x}{2}) = \sqrt{\frac{2 + \sqrt{2 - \sqrt{2}}}{2 \cdot 2}}$

Step 1.5.2

Multiply $2$ by $2$ .

$\cos (\frac{x}{2}) = \sqrt{\frac{2 + \sqrt{2 - \sqrt{2}}}{4}}$

Step 1.6

Rewrite $\sqrt{\frac{2 + \sqrt{2 - \sqrt{2}}}{4}}$ as $\frac{\sqrt{2 + \sqrt{2 - \sqrt{2}}}}{\sqrt{4}}$ .

$\cos (\frac{x}{2}) = \frac{\sqrt{2 + \sqrt{2 - \sqrt{2}}}}{\sqrt{4}}$

Step 1.7

Simplify the denominator.

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Step 1.7.1

Rewrite $4$ as $2^{2}$ .

$\cos (\frac{x}{2}) = \frac{\sqrt{2 + \sqrt{2 - \sqrt{2}}}}{\sqrt{2^{2}}}$

Step 1.7.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 2

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

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

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (\frac{\sqrt{2 + \sqrt{2 - \sqrt{2}}}}{2})$ .

$\frac{x}{2} = 33.75$

Step 4

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \cdot 33.75$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \cdot 33.75$

Step 5.1.1.2

Rewrite the expression.

$x = 2 \cdot 33.75$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $2$ by $33.75$ .

$x = 67.5$

Step 6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the fourth quadrant.

$\frac{x}{2} = 360 - 33.75$

Step 7

Solve for $x$ .

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Step 7.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (360 - 33.75)$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (360 - 33.75)$

Step 7.2.1.1.2

Rewrite the expression.

$x = 2 (360 - 33.75)$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $2 (360 - 33.75)$ .

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Step 7.2.2.1.1

Subtract $33.75$ from $360$ .

$x = 2 \cdot 326.25$

Step 7.2.2.1.2

Multiply $2$ by $326.25$ .

$x = 652.5$

Step 8

Find the period of $\cos (\frac{x}{2})$ .

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Step 8.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

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

Step 8.2

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

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

Step 8.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

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

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$360 \cdot 2$

Step 8.5

Multiply $360$ by $2$ .

$720$

Step 9

The period of the $\cos (\frac{x}{2})$ function is $720$ so values will repeat every $720$ degrees in both directions.

$x = 67.5 + 720 n, 652.5 + 720 n$ , for any integer $n$