Finite Math Examples

$\sin (165)$

Step 1

The complement of $\sin (165)$ is the angle that when added to $\sin (165)$ forms a right angle ( $90 °$ ).

$90 ° - \sin (165)$

Step 2

The exact value of $\sin (165)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

Step 2.2

Split $15$ into two angles where the values of the six trigonometric functions are known.

Step 2.3

Separate negation.

Step 2.4

Apply the difference of angles identity.

Step 2.5

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

Step 2.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

Step 2.7

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

Step 2.8

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

Step 2.9

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 2.9.1

Simplify each term.

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Step 2.9.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 2.9.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

Step 2.9.1.1.2

Combine using the product rule for radicals.

Step 2.9.1.1.3

Multiply $2$ by $3$ .

Step 2.9.1.1.4

Multiply $2$ by $2$ .

Step 2.9.1.2

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

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Step 2.9.1.2.1

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

Step 2.9.1.2.2

Multiply $2$ by $2$ .

Step 2.9.2

Combine the numerators over the common denominator.

$90 - \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 3

To write $90$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$90 \cdot \frac{4}{4} - \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 4

Combine fractions.

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Step 4.1

Combine $90$ and $\frac{4}{4}$ .

$\frac{90 \cdot 4}{4} - \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{90 \cdot 4 - (\sqrt{6} - \sqrt{2})}{4 °}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $90$ by $4$ .

$\frac{360 - (\sqrt{6} - \sqrt{2})}{4 °}$

Step 5.2

Apply the distributive property.

$\frac{360 - \sqrt{6} + \sqrt{2}}{4 °}$

Step 5.3

Multiply $- - \sqrt{2}$ .

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Step 5.3.1

Multiply $- 1$ by $- 1$ .

$\frac{360 - \sqrt{6} + 1 \sqrt{2}}{4 °}$

Step 5.3.2

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

$\frac{360 - \sqrt{6} + \sqrt{2}}{4 °}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{360 - \sqrt{6} + \sqrt{2}}{4 °}$

Decimal Form:

$89.74118095 \dots$