Trigonometry Examples

$50 \cos (75)$

Step 1

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

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Step 1.1

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

$50 \cos (30 + 45)$

Step 1.2

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

$50 (\cos (30) \cos (45) - \sin (30) \sin (45))$

Step 1.3

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

$50 (\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45))$

Step 1.4

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

$50 (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45))$

Step 1.5

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

$50 (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45))$

Step 1.6

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

$50 (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 1.7

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

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

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

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Step 1.7.1.1.1

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

$50 (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 1.7.1.1.2

Combine using the product rule for radicals.

$50 (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 1.7.1.1.3

Multiply $3$ by $2$ .

$50 (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 1.7.1.1.4

Multiply $2$ by $2$ .

$50 (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 1.7.1.2

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

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Step 1.7.1.2.1

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

$50 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Step 1.7.1.2.2

Multiply $2$ by $2$ .

$50 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Step 1.7.2

Combine the numerators over the common denominator.

$50 \frac{\sqrt{6} - \sqrt{2}}{4}$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Factor $2$ out of $50$ .

$2 (25) \frac{\sqrt{6} - \sqrt{2}}{4}$

Step 2.2

Factor $2$ out of $4$ .

$2 \cdot 25 \frac{\sqrt{6} - \sqrt{2}}{2 \cdot 2}$

Step 2.3

Cancel the common factor.

$2 \cdot 25 \frac{\sqrt{6} - \sqrt{2}}{2 \cdot 2}$

Step 2.4

Rewrite the expression.

$25 \frac{\sqrt{6} - \sqrt{2}}{2}$

Step 3

Combine $25$ and $\frac{\sqrt{6} - \sqrt{2}}{2}$ .

$\frac{25 (\sqrt{6} - \sqrt{2})}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{25 (\sqrt{6} - \sqrt{2})}{2}$

Decimal Form:

$12.94095225 \dots$