1. Let Z -3+2i find r and q
Draw the angle Q and leave your answer in polar form
To find the modulus (r) and argument (q) of the complex number Z = -3 + 2i, we can use the formulas:
r = sqrt(a^2 + b^2)
q = arctan(b/a)
where a is the real part and b is the imaginary part.
In this case, a = -3 and b = 2.
r = sqrt((-3)^2 + 2^2) = sqrt(9 + 4) = sqrt(13)
To find q, we can use the arctan function. However, it is important to note that since the real part (a) is negative, we need to add π (180 degrees) to the result of the arctan function.
q = arctan(2/(-3)) + π
Using a calculator or reference table, arctan(2/(-3)) is approximately -0.588 radians or -33.69 degrees.
Therefore, q ≈ -0.588 + π radians.
The polar form of Z is therefore:
Z = r(cos(q) + isin(q))
= sqrt(13)(cos(-0.588 + π) + isin(-0.588 + π))
Note: The value of π in the polar form can be adjusted depending on the range required.