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

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

Evaluate

$$\frac{1}{3}\approx 0.333333333$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$4\times \frac{1}{8}-\frac{2}{3}\left(\left(\sin(60)\right)^{2}-\left(\cos(45)\right)^{2}\right)$$

Multiply 4 and \frac{1}{8} to get \frac{1}{2}.

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

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

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

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

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

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

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

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

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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}.

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

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

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

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

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

Cancel out 2 in both numerator and denominator.

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

The square of \sqrt{3} is 3.

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

Subtract 2 from 3 to get 1.

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

Multiply 2 and 3 to get 6.

$$\frac{1}{2}-\frac{1}{6}$$

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

$$\frac{1}{3}$$