Solve 3sqrt{2}sin45^circ-3(tan45^circ-cos30^circ)

$$3 \sqrt { 2 } \sin 45 ^ { \circ } - 3 ( \tan 45 ^ { \circ } - \cos 30 ^ { \circ } )$$

Evaluate

$$\frac{3\sqrt{3}}{2}\approx 2.598076211$$

Short Solution Steps

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

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

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

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

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

$$\frac{3\sqrt{2}}{2}\sqrt{2}-3\left(1-\cos(30)\right)$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

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

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

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

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

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

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

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

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

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

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

Do the multiplications in 3\sqrt{2}\sqrt{2}-\left(6-3\sqrt{3}\right).

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

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

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