Precalculus Examples

$\cos (75)$

Step 1

The complement of $\cos (75)$ is the angle that when added to $\cos (75)$ forms a right angle ( $90 °$ ).

$90 ° - \cos (75)$

Step 2

The exact value of $\cos (75)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.1

Split $75$ into two angles where the values of the six trigonometric functions are known.

Step 2.2

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

Step 2.3

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

Step 2.4

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

Step 2.5

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

Step 2.6

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

Step 2.7

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

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

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

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Step 2.7.1.1.1

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

Step 2.7.1.1.2

Combine using the product rule for radicals.

Step 2.7.1.1.3

Multiply $3$ by $2$ .

Step 2.7.1.1.4

Multiply $2$ by $2$ .

Step 2.7.1.2

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

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Step 2.7.1.2.1

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

Step 2.7.1.2.2

Multiply $2$ by $2$ .

Step 2.7.2

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:

$89.74118095 \dots$