Trigonometry Examples

$\sin (30) + \cos (30) + \tan (30)$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{1}{2} + \cos (30) + \tan (30)$

Step 1.2

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

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

Step 1.3

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

$\frac{1}{2} + \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{3}$

Step 2

To write $\frac{\sqrt{3}}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3

To write $\frac{\sqrt{3}}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Multiply $2$ by $3$ .

$\frac{1}{2} + \frac{\sqrt{3} \cdot 3}{6} + \frac{\sqrt{3}}{3} \cdot \frac{2}{2}$

Step 4.3

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

$\frac{1}{2} + \frac{\sqrt{3} \cdot 3}{6} + \frac{\sqrt{3} \cdot 2}{3 \cdot 2}$

Step 4.4

Multiply $3$ by $2$ .

$\frac{1}{2} + \frac{\sqrt{3} \cdot 3}{6} + \frac{\sqrt{3} \cdot 2}{6}$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{2} + \frac{\sqrt{3} \cdot 3 + \sqrt{3} \cdot 2}{6}$

Step 6

Simplify the numerator.

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Step 6.1

Move $3$ to the left of $\sqrt{3}$ .

$\frac{1}{2} + \frac{3 \cdot \sqrt{3} + \sqrt{3} \cdot 2}{6}$

Step 6.2

Move $2$ to the left of $\sqrt{3}$ .

$\frac{1}{2} + \frac{3 \sqrt{3} + 2 \cdot \sqrt{3}}{6}$

Step 6.3

Add $3 \sqrt{3}$ and $2 \sqrt{3}$ .

$\frac{1}{2} + \frac{5 \sqrt{3}}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2} + \frac{5 \sqrt{3}}{6}$

Decimal Form:

$1.94337567 \dots$