Trigonometry Examples

$\sin (67.5 °) \cos (22.5 °)$

Step 1

The exact value of $\sin (67.5 °)$ 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 $2$ .

$\sin (\frac{135}{2}) \cos (22.5 °)$

Step 1.2

Apply the sine half-angle identity.

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

Step 1.3

Change the $\pm$ to $+$ because sine 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

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

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Step 1.4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.4.3.2

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

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

Step 1.4.4

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

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

Step 1.4.5

Combine the numerators over the common denominator.

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

Step 1.4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.4.7

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

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Step 1.4.7.1

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

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

Step 1.4.7.2

Multiply $2$ by $2$ .

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

Step 1.4.8

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 1.4.9

Simplify the denominator.

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Step 1.4.9.1

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

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

Step 1.4.9.2

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

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Multiply $2$ by $2$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Simplify.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.85355339 \dots$