what is z= -2-2square root of 3i in trigonometric form.
-2 - 2√3 i corresponds to the point (-2,-2√3)
so, tanθ = y/x = -2√3/-2 = √3
that puts θ in QIII, so θ = 4pi/3
r^2 = 4+12 = 16, so r = 4
so, -2 - 2√3 i = (4,4pi/3)
To represent a complex number in its trigonometric form, we can use the following formula:
z = r(cosθ + isinθ),
where r is the magnitude of the complex number and θ is its argument.
To find the magnitude (r), we can use the formula:
|r| = √(a^2 + b^2),
where a is the real part of the complex number and b is the imaginary part.
In this case, the complex number is z = -2 - 2√3i.
The real part (a) is -2, and the imaginary part (b) is -2√3.
|r| = √((-2)^2 + (-2√3)^2) = √(4 + 12) = √16 = 4.
Now, to find the argument (θ), we can use the formula:
θ = atan(b/a),
where atan is the arctangent function.
θ = atan((-2√3) / (-2)) = atan(√3) ≈ 60°.
Therefore, the trigonometric form of z is:
z = 4(cos60° + isin60°).