Solve frac{sin(60^{circ})cos(30^circ)-tan(30)}{tan(45^circ)-cos(60^circ)sin(30^circ)}

$$\frac{ \sin ( 60 ^ { \circ } ) \cos ( 30 ^ { \circ } ) - \tan ( 30 ) }{ \tan ( 45 ^ { \circ } ) - \cos ( 60 ^ { \circ } ) \sin ( 30 ^ { \circ } ) }$$

Evaluate

$$-\frac{4\sqrt{3}}{9}+1\approx 0.230199641$$

Short Solution Steps

Get the value of \sin(60) from trigonometric values table.

$$\frac{\frac{\sqrt{3}}{2}\cos(30)-\tan(30)}{\tan(45)-\cos(60)\sin(30)}$$

Get the value of \cos(30) from trigonometric values table.

$$\frac{\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2}-\tan(30)}{\tan(45)-\cos(60)\sin(30)}$$

Multiply \frac{\sqrt{3}}{2} and \frac{\sqrt{3}}{2} to get \left(\frac{\sqrt{3}}{2}\right)^{2}.

$$\frac{\left(\frac{\sqrt{3}}{2}\right)^{2}-\tan(30)}{\tan(45)-\cos(60)\sin(30)}$$

To raise \frac{\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\tan(30)}{\tan(45)-\cos(60)\sin(30)}$$

Get the value of \tan(30) from trigonometric values table.

$$\frac{\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\frac{\sqrt{3}}{3}}{\tan(45)-\cos(60)\sin(30)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2} and 3 is 12. Multiply \frac{\left(\sqrt{3}\right)^{2}}{2^{2}} times \frac{3}{3}. Multiply \frac{\sqrt{3}}{3} times \frac{4}{4}.

$$\frac{\frac{3\left(\sqrt{3}\right)^{2}}{12}-\frac{4\sqrt{3}}{12}}{\tan(45)-\cos(60)\sin(30)}$$

Since \frac{3\left(\sqrt{3}\right)^{2}}{12} and \frac{4\sqrt{3}}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3\left(\sqrt{3}\right)^{2}-4\sqrt{3}}{12}}{\tan(45)-\cos(60)\sin(30)}$$

Get the value of \tan(45) from trigonometric values table.

$$\frac{\frac{3\left(\sqrt{3}\right)^{2}-4\sqrt{3}}{12}}{1-\cos(60)\sin(30)}$$

Get the value of \cos(60) from trigonometric values table.

$$\frac{\frac{3\left(\sqrt{3}\right)^{2}-4\sqrt{3}}{12}}{1-\frac{1}{2}\sin(30)}$$

Get the value of \sin(30) from trigonometric values table.

$$\frac{\frac{3\left(\sqrt{3}\right)^{2}-4\sqrt{3}}{12}}{1-\frac{1}{2}\times \frac{1}{2}}$$

Multiply \frac{1}{2} and \frac{1}{2} to get \frac{1}{4}.

$$\frac{\frac{3\left(\sqrt{3}\right)^{2}-4\sqrt{3}}{12}}{1-\frac{1}{4}}$$

Subtract \frac{1}{4} from 1 to get \frac{3}{4}.

$$\frac{\frac{3\left(\sqrt{3}\right)^{2}-4\sqrt{3}}{12}}{\frac{3}{4}}$$

Divide \frac{3\left(\sqrt{3}\right)^{2}-4\sqrt{3}}{12} by \frac{3}{4} by multiplying \frac{3\left(\sqrt{3}\right)^{2}-4\sqrt{3}}{12} by the reciprocal of \frac{3}{4}.

$$\frac{\left(3\left(\sqrt{3}\right)^{2}-4\sqrt{3}\right)\times 4}{12\times 3}$$

Cancel out 4 in both numerator and denominator.

$$\frac{-4\sqrt{3}+3\left(\sqrt{3}\right)^{2}}{3\times 3}$$

The square of \sqrt{3} is 3.

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

Multiply 3 and 3 to get 9.

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

Multiply 3 and 3 to get 9.

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