Solve sqrt{2}sin45^circ-6tan30^circ+2cos30^circ

$$\sqrt { 2 } \sin 45 ^ { \circ } - 6 \tan 30 ^ { \circ } + 2 \cos 30 ^ { \circ }$$

Evaluate

$$1-\sqrt{3}\approx -0.732050808$$

Short Solution Steps

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

$$\sqrt{2}\times \frac{\sqrt{2}}{2}-6\tan(30)+2\cos(30)$$

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

$$\frac{\sqrt{2}\sqrt{2}}{2}-6\tan(30)+2\cos(30)$$

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

$$\frac{\sqrt{2}\sqrt{2}}{2}-6\times \frac{\sqrt{3}}{3}+2\cos(30)$$

Cancel out 3, the greatest common factor in 6 and 3.

$$\frac{\sqrt{2}\sqrt{2}}{2}-2\sqrt{3}+2\cos(30)$$

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

$$\frac{\sqrt{2}\sqrt{2}}{2}+\frac{2\left(-2\right)\sqrt{3}}{2}+2\cos(30)$$

Since \frac{\sqrt{2}\sqrt{2}}{2} and \frac{2\left(-2\right)\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{2}\sqrt{2}+2\left(-2\right)\sqrt{3}}{2}+2\cos(30)$$

Do the multiplications in \sqrt{2}\sqrt{2}+2\left(-2\right)\sqrt{3}.

$$\frac{2-4\sqrt{3}}{2}+2\cos(30)$$

Divide each term of 2-4\sqrt{3} by 2 to get 1-2\sqrt{3}.

$$1-2\sqrt{3}+2\cos(30)$$

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

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

Cancel out 2 and 2.

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

Combine -2\sqrt{3} and \sqrt{3} to get -\sqrt{3}.

$$1-\sqrt{3}$$