Trigonometry Examples

$14 \cdot \frac{\sin (105)}{18}$

Step 1

The exact value of $\sin (105)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 1.1

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

$14 \cdot \frac{\sin (75)}{18}$

Step 1.2

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

$14 \cdot \frac{\sin (30 + 45)}{18}$

Step 1.3

Apply the sum of angles identity.

$14 \cdot \frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{18}$

Step 1.4

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

$14 \cdot \frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{18}$

Step 1.5

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

$14 \cdot \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{18}$

Step 1.6

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

$14 \cdot \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{18}$

Step 1.7

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

$14 \cdot \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{18}$

Step 1.8

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

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

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

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Step 1.8.1.1.1

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

$14 \cdot \frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{18}$

Step 1.8.1.1.2

Multiply $2$ by $2$ .

$14 \cdot \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{18}$

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

$14 \cdot \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{18}$

Step 1.8.1.2.2

Combine using the product rule for radicals.

$14 \cdot \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{18}$

Step 1.8.1.2.3

Multiply $3$ by $2$ .

$14 \cdot \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{18}$

Step 1.8.1.2.4

Multiply $2$ by $2$ .

$14 \cdot \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{18}$

Step 1.8.2

Combine the numerators over the common denominator.

$14 \cdot \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{18}$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Factor $2$ out of $14$ .

$2 (7) \cdot \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{18}$

Step 2.2

Factor $2$ out of $18$ .

$2 \cdot 7 \cdot \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{2 \cdot 9}$

Step 2.3

Cancel the common factor.

$2 \cdot 7 \cdot \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{2 \cdot 9}$

Step 2.4

Rewrite the expression.

$7 \cdot \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{9}$

Step 3

Combine $7$ and $\frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{9}$ .

$\frac{7 \frac{\sqrt{2} + \sqrt{6}}{4}}{9}$

Step 4

Combine $7$ and $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

$\frac{\frac{7 (\sqrt{2} + \sqrt{6})}{4}}{9}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{7 (\sqrt{2} + \sqrt{6})}{4} \cdot \frac{1}{9}$

Step 6

Multiply $\frac{7 (\sqrt{2} + \sqrt{6})}{4} \cdot \frac{1}{9}$ .

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Step 6.1

Multiply $\frac{7 (\sqrt{2} + \sqrt{6})}{4}$ by $\frac{1}{9}$ .

$\frac{7 (\sqrt{2} + \sqrt{6})}{4 \cdot 9}$

Step 6.2

Multiply $4$ by $9$ .

$\frac{7 (\sqrt{2} + \sqrt{6})}{36}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{7 (\sqrt{2} + \sqrt{6})}{36}$

Decimal Form:

$0.75127564 \dots$