Trigonometry Examples

$\cos (15) \cdot 30$

Step 1

The exact value of $\cos (15)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.1

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

$\cos (45 - 30) \cdot 30$

Step 1.2

Separate negation.

$\cos (45 - (30)) \cdot 30$

Step 1.3

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

$(\cos (45) \cos (30) + \sin (45) \sin (30)) \cdot 30$

Step 1.4

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

$(\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) \cdot 30$

Step 1.5

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

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

Step 1.6

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) \cdot 30$

Step 1.7

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

$(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot 30$

Step 1.8

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

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

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

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Step 1.8.1.1.1

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

$(\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot 30$

Step 1.8.1.1.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot 30$

Step 1.8.1.1.3

Multiply $2$ by $3$ .

$(\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot 30$

Step 1.8.1.1.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cdot 30$

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) \cdot 30$

Step 1.8.1.2.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) \cdot 30$

Step 1.8.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} + \sqrt{2}}{4} \cdot 30$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Factor $2$ out of $4$ .

$\frac{\sqrt{6} + \sqrt{2}}{2 (2)} \cdot 30$

Step 2.2

Factor $2$ out of $30$ .

$\frac{\sqrt{6} + \sqrt{2}}{2 \cdot 2} \cdot (2 \cdot 15)$

Step 2.3

Cancel the common factor.

$\frac{\sqrt{6} + \sqrt{2}}{2 \cdot 2} \cdot (2 \cdot 15)$

Step 2.4

Rewrite the expression.

$\frac{\sqrt{6} + \sqrt{2}}{2} \cdot 15$

Step 3

Combine $\frac{\sqrt{6} + \sqrt{2}}{2}$ and $15$ .

$\frac{(\sqrt{6} + \sqrt{2}) \cdot 15}{2}$

Step 4

Move $15$ to the left of $\sqrt{6} + \sqrt{2}$ .

$\frac{15 (\sqrt{6} + \sqrt{2})}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{15 (\sqrt{6} + \sqrt{2})}{2}$

Decimal Form:

$28.97777478 \dots$