Trigonometry Examples

$2 \cos^{2} (22.5) - 1$

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 $2$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

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.

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

Step 1.5.2

Combine the numerators over the common denominator.

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

Step 1.5.3

Multiply the numerator by the reciprocal of the denominator.

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

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 $\frac{1}{2}$ .

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

Step 1.5.4.2

Multiply $2$ by $2$ .

$2 {\sqrt{\frac{2 + \sqrt{2}}{4}}}^{2} - 1$

Step 1.5.5

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

$2 {(\frac{\sqrt{2 + \sqrt{2}}}{\sqrt{4}})}^{2} - 1$

Step 1.5.6

Simplify the denominator.

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Step 1.5.6.1

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

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

Step 1.5.6.2

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

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

Step 2

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

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

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Simplify.

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

Step 4

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

$2 \frac{2 + \sqrt{2}}{4} - 1$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Factor $2$ out of $4$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Rewrite $\frac{2 + \sqrt{2} - 1 \cdot 2}{2}$ in a factored form.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $2$ .

$\frac{0 + \sqrt{2}}{2}$

Step 9.3

Add $0$ and $\sqrt{2}$ .

$\frac{\sqrt{2}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$