Solve 44(sin^{4}30^{circ}+cos^{4}60^{circ})-2/3(sin^{2}60^{circ}-cos^245^circ)+1/2tan^260^circ

$$44 ( \sin ^ { 4 } 30 ^ { \circ } + \cos ^ { 4 } 60 ^ { \circ } ) - \frac { 2 } { 3 } ( \sin ^ { 2 } 60 ^ { \circ } - \cos ^ { 2 } 45 ^ { \circ } ) + \frac { 1 } { 2 } \tan ^ { 2 } 60 ^ { \circ }$$

Evaluate

$$\frac{41}{6}\approx 6.833333333$$

Solution Steps

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

$$44\left(\left(\frac{1}{2}\right)^{4}+\left(\cos(60)\right)^{4}\right)-\frac{2}{3}\left(\left(\sin(60)\right)^{2}-\left(\cos(45)\right)^{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Calculate \frac{1}{2} to the power of 4 and get \frac{1}{16}.

$$44\left(\frac{1}{16}+\left(\cos(60)\right)^{4}\right)-\frac{2}{3}\left(\left(\sin(60)\right)^{2}-\left(\cos(45)\right)^{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

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

$$44\left(\frac{1}{16}+\left(\frac{1}{2}\right)^{4}\right)-\frac{2}{3}\left(\left(\sin(60)\right)^{2}-\left(\cos(45)\right)^{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Calculate \frac{1}{2} to the power of 4 and get \frac{1}{16}.

$$44\left(\frac{1}{16}+\frac{1}{16}\right)-\frac{2}{3}\left(\left(\sin(60)\right)^{2}-\left(\cos(45)\right)^{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Add \frac{1}{16} and \frac{1}{16} to get \frac{1}{8}.

$$44\times \frac{1}{8}-\frac{2}{3}\left(\left(\sin(60)\right)^{2}-\left(\cos(45)\right)^{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Multiply 44 and \frac{1}{8} to get \frac{11}{2}.

$$\frac{11}{2}-\frac{2}{3}\left(\left(\sin(60)\right)^{2}-\left(\cos(45)\right)^{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

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

$$\frac{11}{2}-\frac{2}{3}\left(\left(\frac{\sqrt{3}}{2}\right)^{2}-\left(\cos(45)\right)^{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

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

$$\frac{11}{2}-\frac{2}{3}\left(\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\left(\cos(45)\right)^{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

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

$$\frac{11}{2}-\frac{2}{3}\left(\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\left(\frac{\sqrt{2}}{2}\right)^{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

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

$$\frac{11}{2}-\frac{2}{3}\left(\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

The square of \sqrt{2} is 2.

$$\frac{11}{2}-\frac{2}{3}\left(\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\frac{2}{2^{2}}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{11}{2}-\frac{2}{3}\left(\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\frac{2}{4}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{11}{2}-\frac{2}{3}\left(\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\frac{1}{2}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

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{1}{2} times \frac{2}{2}.

$$\frac{11}{2}-\frac{2}{3}\left(\frac{\left(\sqrt{3}\right)^{2}}{4}-\frac{2}{4}\right)+\frac{1}{2}\left(\tan(60)\right)^{2}$$

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

$$\frac{11}{2}-\frac{2}{3}\times \frac{\left(\sqrt{3}\right)^{2}-2}{4}+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Multiply \frac{2}{3} times \frac{\left(\sqrt{3}\right)^{2}-2}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{11}{2}-\frac{2\left(\left(\sqrt{3}\right)^{2}-2\right)}{3\times 4}+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{11}{2}-\frac{\left(\sqrt{3}\right)^{2}-2}{2\times 3}+\frac{1}{2}\left(\tan(60)\right)^{2}$$

The square of \sqrt{3} is 3.

$$\frac{11}{2}-\frac{3-2}{2\times 3}+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Subtract 2 from 3 to get 1.

$$\frac{11}{2}-\frac{1}{2\times 3}+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Multiply 2 and 3 to get 6.

$$\frac{11}{2}-\frac{1}{6}+\frac{1}{2}\left(\tan(60)\right)^{2}$$

Subtract \frac{1}{6} from \frac{11}{2} to get \frac{16}{3}.

$$\frac{16}{3}+\frac{1}{2}\left(\tan(60)\right)^{2}$$

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

$$\frac{16}{3}+\frac{1}{2}\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\frac{16}{3}+\frac{1}{2}\times 3$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{16}{3}+\frac{3}{2}$$

Add \frac{16}{3} and \frac{3}{2} to get \frac{41}{6}.

$$\frac{41}{6}$$