Trigonometry Examples

$\cos (67.5) \cos (22.5)$

Step 1

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

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Step 1.1

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

$\cos (\frac{135}{2}) \cos (22.5)$

Step 1.2

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

$(\pm \sqrt{\frac{1 + \cos (135)}{2}}) \cos (22.5)$

Step 1.3

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

$\sqrt{\frac{1 + \cos (135)}{2}} \cos (22.5)$

Step 1.4

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

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Step 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.

$\sqrt{\frac{1 - \cos (45)}{2}} \cos (22.5)$

Step 1.4.2

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

$\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \cos (22.5)$

Step 1.4.3

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

$\sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}} \cos (22.5)$

Step 1.4.4

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} \cos (22.5)$

Step 1.4.5

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}} \cos (22.5)$

Step 1.4.6

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

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Step 1.4.6.1

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

$\sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} \cos (22.5)$

Step 1.4.6.2

Multiply $2$ by $2$ .

$\sqrt{\frac{2 - \sqrt{2}}{4}} \cos (22.5)$

Step 1.4.7

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

$\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}} \cos (22.5)$

Step 1.4.8

Simplify the denominator.

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Step 1.4.8.1

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

$\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}} \cos (22.5)$

Step 1.4.8.2

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

$\frac{\sqrt{2 - \sqrt{2}}}{2} \cos (22.5)$

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Combine the numerators over the common denominator.

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

Step 2.5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.4

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

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Step 2.5.4.1

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

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

Step 2.5.4.2

Multiply $2$ by $2$ .

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

Step 2.5.5

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

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

Step 2.5.6

Simplify the denominator.

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Step 2.5.6.1

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

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

Step 2.5.6.2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Combine using the product rule for radicals.

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

Step 3.3

Expand $(2 - \sqrt{2}) (2 + \sqrt{2})$ using the FOIL Method.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Apply the distributive property.

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

Step 3.4

Simplify and combine like terms.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $2$ by $2$ .

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

Step 3.4.1.2

Multiply $2$ by $- 1$ .

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

Step 3.4.1.3

Multiply $- \sqrt{2} \sqrt{2}$ .

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Step 3.4.1.3.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.4.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.4.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4.1.3.4

Add $1$ and $1$ .

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

Step 3.4.1.4

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

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Step 3.4.1.4.1

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

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

Step 3.4.1.4.2

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

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

Step 3.4.1.4.3

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

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

Step 3.4.1.4.4

Cancel the common factor of $2$ .

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Step 3.4.1.4.4.1

Cancel the common factor.

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

Step 3.4.1.4.4.2

Rewrite the expression.

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

Step 3.4.1.4.5

Evaluate the exponent.

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

Step 3.4.1.5

Multiply $- 1$ by $2$ .

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

Step 3.4.2

Subtract $2$ from $4$ .

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

Step 3.4.3

Subtract $2 \sqrt{2}$ from $2 \sqrt{2}$ .

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

Step 3.4.4

Add $2$ and $0$ .

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

Step 3.5

Multiply $2$ by $2$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.35355339 \dots$