Solve 5-tan(pi/3)/sin(frac{pi{3})}

$$\frac{ 5- \tan ( \frac{ \pi }{ 3 } ) }{ \sin ( \frac{ \pi }{ 3 } ) }$$

Evaluate

$$\frac{10\sqrt{3}}{3}-2\approx 3.773502692$$

Solution Steps

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

$$\frac{5-\sqrt{3}}{\sin(\frac{\pi }{3})}$$

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

$$\frac{5-\sqrt{3}}{\frac{\sqrt{3}}{2}}$$

Divide 5-\sqrt{3} by \frac{\sqrt{3}}{2} by multiplying 5-\sqrt{3} by the reciprocal of \frac{\sqrt{3}}{2}.

$$\frac{\left(5-\sqrt{3}\right)\times 2}{\sqrt{3}}$$

Rationalize the denominator of \frac{\left(5-\sqrt{3}\right)\times 2}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(5-\sqrt{3}\right)\times 2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(5-\sqrt{3}\right)\times 2\sqrt{3}}{3}$$

Use the distributive property to multiply 5-\sqrt{3} by 2.

$$\frac{\left(10-2\sqrt{3}\right)\sqrt{3}}{3}$$

Use the distributive property to multiply 10-2\sqrt{3} by \sqrt{3}.

$$\frac{10\sqrt{3}-2\left(\sqrt{3}\right)^{2}}{3}$$

The square of \sqrt{3} is 3.

$$\frac{10\sqrt{3}-2\times 3}{3}$$

Multiply -2 and 3 to get -6.

$$\frac{10\sqrt{3}-6}{3}$$