Solve 2cospi/3cospi/3+(-2sinpi/3)sinfrac{pi/3}{-2cosfrac{pi}{3}sinfrac{pi}{3}+(-2sinfrac{pi}{3})cosfrac{pi}{3}}=

$$\frac { 2 \cos \frac { \pi } { 3 } \cos \frac { \pi } { 3 } + ( - 2 \sin \frac { \pi } { 3 } ) \sin \frac { \pi } { 3 } } { - 2 \cos \frac { \pi } { 3 } \sin \frac { \pi } { 3 } + ( - 2 \sin \frac { \pi } { 3 } ) \cos \frac { \pi } { 3 } } =$$

Evaluate

$$\frac{\sqrt{3}}{3}\approx 0.577350269$$

Solution Steps

Multiply \cos(\frac{\pi }{3}) and \cos(\frac{\pi }{3}) to get \left(\cos(\frac{\pi }{3})\right)^{2}.

$$\frac{2\left(\cos(\frac{\pi }{3})\right)^{2}-2\sin(\frac{\pi }{3})\sin(\frac{\pi }{3})}{-2\cos(\frac{\pi }{3})\sin(\frac{\pi }{3})-2\sin(\frac{\pi }{3})\cos(\frac{\pi }{3})}$$

Multiply \sin(\frac{\pi }{3}) and \sin(\frac{\pi }{3}) to get \left(\sin(\frac{\pi }{3})\right)^{2}.

$$\frac{2\left(\cos(\frac{\pi }{3})\right)^{2}-2\left(\sin(\frac{\pi }{3})\right)^{2}}{-2\cos(\frac{\pi }{3})\sin(\frac{\pi }{3})-2\sin(\frac{\pi }{3})\cos(\frac{\pi }{3})}$$

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

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

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

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

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

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

Get the value of \sin(\frac{\pi }{3}) from trigonometric values table.

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

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

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

Express -2\times \frac{\left(\sqrt{3}\right)^{2}}{2^{2}} as a single fraction.

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

Cancel out 2 in both numerator and denominator.

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

Since \frac{1}{2} and \frac{-\left(\sqrt{3}\right)^{2}}{2} have the same denominator, add them by adding their numerators.

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

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

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

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

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

Get the value of \sin(\frac{\pi }{3}) from trigonometric values table.

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

Get the value of \sin(\frac{\pi }{3}) from trigonometric values table.

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

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

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

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

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

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

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

Cancel out 2 and 2.

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

Express \frac{\frac{1-\left(\sqrt{3}\right)^{2}}{2}}{-\sqrt{3}} as a single fraction.

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

Rationalize the denominator of \frac{1-\left(\sqrt{3}\right)^{2}}{2\left(-1\right)\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

The square of \sqrt{3} is 3.

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

Subtract 3 from 1 to get -2.

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

Multiply 2 and -1 to get -2.

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

Multiply -2 and 3 to get -6.

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

Divide -2\sqrt{3} by -6 to get \frac{1}{3}\sqrt{3}.

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