Solve 0=z+2cos(pi/6)+isqrt{3}sin(pi/2)

$$0 = z + 2 \cos ( \frac { \pi } { 6 } ) + i \sqrt { 3 } \sin ( \frac { \pi } { 2 } )$$

Solve for z

$$z=\sqrt{3}\left(-1-i\right)\approx -1.732050808-1.732050808i$$

Steps for Solving Linear Equation

Get the value of \cos(\frac{\pi }{6}) from trigonometric values table.

$$0=z+2\times \frac{\sqrt{3}}{2}+i\sqrt{3}\sin(\frac{\pi }{2})$$

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

$$0=z+\frac{2\sqrt{3}}{2}+i\sqrt{3}\sin(\frac{\pi }{2})$$

Cancel out 2 and 2.

$$0=z+\sqrt{3}+i\sqrt{3}\sin(\frac{\pi }{2})$$

Get the value of \sin(\frac{\pi }{2}) from trigonometric values table.

$$0=z+\sqrt{3}+i\sqrt{3}\times 1$$

Multiply i and 1 to get i.

$$0=z+\sqrt{3}+i\sqrt{3}$$

Combine \sqrt{3} and i\sqrt{3} to get \left(1+i\right)\sqrt{3}.

$$0=z+\left(1+i\right)\sqrt{3}$$

Swap sides so that all variable terms are on the left hand side.

$$z+\left(1+i\right)\sqrt{3}=0$$

Subtract \left(1+i\right)\sqrt{3} from both sides. Anything subtracted from zero gives its negation.

$$z=\left(-1-i\right)\sqrt{3}$$