Solve 2/tan30^{circ}-2sin60^circcos45^circ+3tan30^circsin45^circ

$$\frac { 2 } { \tan 30 ^ { \circ } } - 2 \sin 60 ^ { \circ } \cos 45 ^ { \circ } + 3 \tan 30 ^ { \circ } \sin 45 ^ { \circ }$$

Evaluate

$$2\sqrt{3}\approx 3.464101615$$

Short Solution Steps

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{3}}{3}-2\sin(60)\cos(45)+3\tan(30)\sin(45)$$

Divide 6\sqrt{3} by 3 to get 2\sqrt{3}.

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

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

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

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

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

Cancel out 2 and 2.

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

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

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$2\sqrt{3}-\frac{\sqrt{6}}{2}+3\tan(30)\sin(45)$$

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

$$\frac{2\times 2\sqrt{3}}{2}-\frac{\sqrt{6}}{2}+3\tan(30)\sin(45)$$

Since \frac{2\times 2\sqrt{3}}{2} and \frac{\sqrt{6}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\times 2\sqrt{3}-\sqrt{6}}{2}+3\tan(30)\sin(45)$$

Do the multiplications in 2\times 2\sqrt{3}-\sqrt{6}.

$$\frac{4\sqrt{3}-\sqrt{6}}{2}+3\tan(30)\sin(45)$$

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

$$\frac{4\sqrt{3}-\sqrt{6}}{2}+3\times \frac{\sqrt{3}}{3}\sin(45)$$

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

$$\frac{4\sqrt{3}-\sqrt{6}}{2}+3\times \frac{\sqrt{3}}{3}\times \frac{\sqrt{2}}{2}$$

Cancel out 3 and 3.

$$\frac{4\sqrt{3}-\sqrt{6}}{2}+\sqrt{3}\times \frac{\sqrt{2}}{2}$$

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

$$\frac{4\sqrt{3}-\sqrt{6}}{2}+\frac{\sqrt{3}\sqrt{2}}{2}$$

Since \frac{4\sqrt{3}-\sqrt{6}}{2} and \frac{\sqrt{3}\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

$$\frac{4\sqrt{3}-\sqrt{6}+\sqrt{3}\sqrt{2}}{2}$$

Do the multiplications in 4\sqrt{3}-\sqrt{6}+\sqrt{3}\sqrt{2}.

$$\frac{4\sqrt{3}-\sqrt{6}+\sqrt{6}}{2}$$

Do the calculations in 4\sqrt{3}-\sqrt{6}+\sqrt{6}.

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

Divide 4\sqrt{3} by 2 to get 2\sqrt{3}.

$$2\sqrt{3}$$