Solve 1/3cos(0^{circ})+sqrt{3}sin(60^{circ})+4cos(90^circ)-sqrt{2/3}cos(45^circ)-2cos(60^circ)-3/2sin(90^circ)

$$\frac{ 1 }{ 3 } \cos ( 0 ^ { \circ } ) + \sqrt{ 3 } \sin ( 60 ^ { \circ } ) +4 \cos ( 90 ^ { \circ } ) - \sqrt{ \frac{ 2 }{ 3 } } \cos ( 45 ^ { \circ } ) -2 \cos ( 60 ^ { \circ } ) - \frac{ 3 }{ 2 } \sin ( 90 ^ { \circ } )$$

Evaluate

$$\frac{-\sqrt{3}-2}{3}\approx -1.244016936$$

Solution Steps

Get the value of \cos(0) from trigonometric values table.

$$\frac{1}{3}\times 1+\sqrt{3}\sin(60)+4\cos(90)-\sqrt{\frac{2}{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

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

$$\frac{1}{3}+\sqrt{3}\sin(60)+4\cos(90)-\sqrt{\frac{2}{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

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

$$\frac{1}{3}+\sqrt{3}\times \frac{\sqrt{3}}{2}+4\cos(90)-\sqrt{\frac{2}{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

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

$$\frac{1}{3}+\frac{\sqrt{3}\sqrt{3}}{2}+4\cos(90)-\sqrt{\frac{2}{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

Get the value of \cos(90) from trigonometric values table.

$$\frac{1}{3}+\frac{\sqrt{3}\sqrt{3}}{2}+4\times 0-\sqrt{\frac{2}{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

Multiply 4 and 0 to get 0.

$$\frac{1}{3}+\frac{\sqrt{3}\sqrt{3}}{2}+0-\sqrt{\frac{2}{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

Add \frac{1}{3} and 0 to get \frac{1}{3}.

$$\frac{1}{3}+\frac{\sqrt{3}\sqrt{3}}{2}-\sqrt{\frac{2}{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

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

$$\frac{1}{3}+\frac{3}{2}-\sqrt{\frac{2}{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

Add \frac{1}{3} and \frac{3}{2} to get \frac{11}{6}.

$$\frac{11}{6}-\sqrt{\frac{2}{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$\frac{11}{6}-\frac{\sqrt{2}}{\sqrt{3}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{11}{6}-\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

The square of \sqrt{3} is 3.

$$\frac{11}{6}-\frac{\sqrt{2}\sqrt{3}}{3}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{11}{6}-\frac{\sqrt{6}}{3}\cos(45)-2\cos(60)-\frac{3}{2}\sin(90)$$

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

$$\frac{11}{6}-\frac{\sqrt{6}}{3}\times \frac{\sqrt{2}}{2}-2\cos(60)-\frac{3}{2}\sin(90)$$

Multiply \frac{\sqrt{6}}{3} times \frac{\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{11}{6}-\frac{\sqrt{6}\sqrt{2}}{3\times 2}-2\cos(60)-\frac{3}{2}\sin(90)$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\frac{11}{6}-\frac{\sqrt{2}\sqrt{3}\sqrt{2}}{3\times 2}-2\cos(60)-\frac{3}{2}\sin(90)$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{11}{6}-\frac{2\sqrt{3}}{3\times 2}-2\cos(60)-\frac{3}{2}\sin(90)$$

Multiply 3 and 2 to get 6.

$$\frac{11}{6}-\frac{2\sqrt{3}}{6}-2\cos(60)-\frac{3}{2}\sin(90)$$

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

$$\frac{11}{6}-\frac{1}{3}\sqrt{3}-2\cos(60)-\frac{3}{2}\sin(90)$$

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

$$\frac{11}{6}-\frac{1}{3}\sqrt{3}-2\times \frac{1}{2}-\frac{3}{2}\sin(90)$$

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

$$\frac{11}{6}-\frac{1}{3}\sqrt{3}-1-\frac{3}{2}\sin(90)$$

Subtract 1 from \frac{11}{6} to get \frac{5}{6}.

$$\frac{5}{6}-\frac{1}{3}\sqrt{3}-\frac{3}{2}\sin(90)$$

Get the value of \sin(90) from trigonometric values table.

$$\frac{5}{6}-\frac{1}{3}\sqrt{3}-\frac{3}{2}\times 1$$

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

$$\frac{5}{6}-\frac{1}{3}\sqrt{3}-\frac{3}{2}$$

Subtract \frac{3}{2} from \frac{5}{6} to get -\frac{2}{3}.

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