Solve {l}{sin^{2}60^{circ}+sin45^circcos45^circ}{-cos30^circtan60^circ}

$$\left. \begin{array} { l } { \sin ^ { 2 } 60 ^ { \circ } + \sin 45 ^ { \circ } \cos 45 ^ { \circ } } \\ { - \cos 30 ^ { \circ } \tan 60 ^ { \circ } } \end{array} \right.$$

Evaluate

$$-\frac{1}{4}=-0.25$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{\left(\sqrt{3}\right)^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{\sqrt{3}}{2}\tan(60)$$

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

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

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

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

The square of \sqrt{2} is 2.

$$\frac{3+2}{2^{2}}-\frac{3}{2}$$

Add 3 and 2 to get 5.

$$\frac{5}{2^{2}}-\frac{3}{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{5}{4}-\frac{3}{2}$$

Subtract \frac{3}{2} from \frac{5}{4} to get -\frac{1}{4}.

$$-\frac{1}{4}$$