Solve 2*(sinpi/3+1/sqrt{3}cospi/3)+3

$$2 \cdot ( \sin \frac { \pi } { 3 } + \frac { 1 } { \sqrt { 3 } } \cos \frac { \pi } { 3 } ) + 3$$

Evaluate

$$\frac{4\sqrt{3}}{3}+3\approx 5.309401077$$

Short Solution Steps

Get the value of \sin(\frac{\pi }{3}) from trigonometric values table.

$$2\left(\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{3}}\cos(\frac{\pi }{3})\right)+3$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$2\left(\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\cos(\frac{\pi }{3})\right)+3$$

The square of \sqrt{3} is 3.

$$2\left(\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{3}\cos(\frac{\pi }{3})\right)+3$$

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

$$2\left(\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{3}\times \frac{1}{2}\right)+3$$

Multiply \frac{\sqrt{3}}{3} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$2\left(\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{3\times 2}\right)+3$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3\times 2 is 2\times 3. Multiply \frac{\sqrt{3}}{2} times \frac{3}{3}.

$$2\left(\frac{3\sqrt{3}}{2\times 3}+\frac{\sqrt{3}}{2\times 3}\right)+3$$

Since \frac{3\sqrt{3}}{2\times 3} and \frac{\sqrt{3}}{2\times 3} have the same denominator, add them by adding their numerators.

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

Do the calculations in 3\sqrt{3}+\sqrt{3}.

$$2\times \frac{4\sqrt{3}}{2\times 3}+3$$

Cancel out 2 in both numerator and denominator.

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

Express 2\times \frac{2\sqrt{3}}{3} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{3}{3}.

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

Since \frac{2\times 2\sqrt{3}}{3} and \frac{3\times 3}{3} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\times 2\sqrt{3}+3\times 3.

$$\frac{4\sqrt{3}+9}{3}$$