Solve 1/sin(frac{pi{3})}-1/sin(frac{pi{4})}

$$\frac{ 1 }{ \sin ( \frac{ \pi }{ 3 } ) } - \frac{ 1 }{ \sin ( \frac{ \pi }{ 4 } ) }$$

Evaluate

$$\frac{\sqrt{2}\left(\sqrt{6}-3\right)}{3}\approx -0.259513024$$

Short Solution Steps

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

$$\frac{1}{\frac{\sqrt{3}}{2}}-\frac{1}{\sin(\frac{\pi }{4})}$$

Divide 1 by \frac{\sqrt{3}}{2} by multiplying 1 by the reciprocal of \frac{\sqrt{3}}{2}.

$$\frac{2}{\sqrt{3}}-\frac{1}{\sin(\frac{\pi }{4})}$$

Rationalize the denominator of \frac{2}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\frac{1}{\sin(\frac{\pi }{4})}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}}{3}-\frac{1}{\sin(\frac{\pi }{4})}$$

Get the value of \sin(\frac{\pi }{4}) from trigonometric values table.

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

Divide 1 by \frac{\sqrt{2}}{2} by multiplying 1 by the reciprocal of \frac{\sqrt{2}}{2}.

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

Rationalize the denominator of \frac{2}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Cancel out 2 and 2.

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

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

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

Since \frac{2\sqrt{3}}{3} and \frac{3\sqrt{2}}{3} have the same denominator, subtract them by subtracting their numerators.

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