Trigonometry Examples

$\cos^{2} (22.5)$

Step 1

The exact value of $\cos (22.5)$ is $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

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Step 1.1

Rewrite $22.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos^{2} (\frac{45}{2})$

Step 1.2

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

${(\pm \sqrt{\frac{1 + \cos (45)}{2}})}^{2}$

Step 1.3

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

${\sqrt{\frac{1 + \cos (45)}{2}}}^{2}$

Step 1.4

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

${\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}}^{2}$

Step 1.5

Simplify $\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$ .

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Step 1.5.1

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

${\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}}}^{2}$

Step 1.5.2

Combine the numerators over the common denominator.

${\sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}}}^{2}$

Step 1.5.3

Multiply the numerator by the reciprocal of the denominator.

${\sqrt{\frac{2 + \sqrt{2}}{2} \cdot \frac{1}{2}}}^{2}$

Step 1.5.4

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

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Step 1.5.4.1

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

${\sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}}}^{2}$

Step 1.5.4.2

Multiply $2$ by $2$ .

${\sqrt{\frac{2 + \sqrt{2}}{4}}}^{2}$

Step 1.5.5

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

${(\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}})}^{2}$

Step 1.5.6

Simplify the denominator.

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Step 1.5.6.1

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

${(\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2^{2}}})}^{2}$

Step 1.5.6.2

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

${(\frac{\sqrt{2 + \sqrt{2}}}{2})}^{2}$

Step 2

Apply the product rule to $\frac{\sqrt{2 + \sqrt{2}}}{2}$ .

$\frac{{\sqrt{2 + \sqrt{2}}}^{2}}{2^{2}}$

Step 3

Rewrite ${\sqrt{2 + \sqrt{2}}}^{2}$ as $2 + \sqrt{2}$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 + \sqrt{2}}$ as ${(2 + \sqrt{2})}^{\frac{1}{2}}$ .

$\frac{{({(2 + \sqrt{2})}^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(2 + \sqrt{2})}^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 3.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{{(2 + \sqrt{2})}^{\frac{2}{2}}}{2^{2}}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{{(2 + \sqrt{2})}^{\frac{2}{2}}}{2^{2}}$

Step 3.4.2

Rewrite the expression.

$\frac{{(2 + \sqrt{2})}^{1}}{2^{2}}$

Step 3.5

Simplify.

$\frac{2 + \sqrt{2}}{2^{2}}$

Step 4

Raise $2$ to the power of $2$ .

$\frac{2 + \sqrt{2}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2 + \sqrt{2}}{4}$

Decimal Form:

$0.85355339 \dots$