How do you find a complex number when given the modulus and argument

How would you do the following question?
Find in the form of x+iy (where x and y are real) the complex number with modulus 1 and argument 2π/3.
Please explain with every step shown, thanks so much in advance!

x^2 + y^2 = 1
tan 2π/3 = y/x
-√3 = y/x
-y√3 = x

(-y√3)^2 + y^2 = 1
3y^2 + y^2 = 1
4y^2 = 1
y^2 = 1/4
y = ± 1/2

x = ± √3/2

-√3/2 + 1/2 i

Ed I's answer is awesome, I just wanted to make sure it is understood that when we say

x^2 + y^2 = 1, there was some simplification, in even rawer form it is x^2 + y^2 = 1^2, so if your modulus was different than 1, say it is 5, your equation would be

e.g. x^2 + y^2 = 5^2

Ed I soundly did not write 1^2 because 1^2 = 1, just wanted to make sure you did not generalize this into future calculations incorrectly. The modulus R is given by x^2 + y^2 = R^2.