Trigonometry Examples

$\sin (75)$

Step 1

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

$90 ° - \sin (75)$

Step 2

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

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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.

Step 2.3

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

Step 2.4

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

Step 2.5

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

Step 2.6

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

Step 2.7

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

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

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

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Step 2.7.1.1.1

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

Step 2.7.1.1.2

Multiply $2$ by $2$ .

Step 2.7.1.2

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

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Step 2.7.1.2.1

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

Step 2.7.1.2.2

Combine using the product rule for radicals.

Step 2.7.1.2.3

Multiply $3$ by $2$ .

Step 2.7.1.2.4

Multiply $2$ by $2$ .

Step 2.7.2

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $90$ by $4$ .

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

Step 5.2

Apply the distributive property.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$89.03407417 \dots$