how to change -2i in polar form
Answers
Answer:
COmplex no.
Step-by-step explanation:
Consider the complex number z as shown on the complex plane below.
z and w drawn on the complex planeDraw a line segment from 0 to z. Then we can figure out the exact position of z on the complex plane if we know two things: the length of the line segment and the angle measured from the positive real axis to the line segment. The length of the line segment is given by | z |, which is (−2)2+12−−−−−−−−−√=5–√. What about the angle?
Consider the picture below. The angle that we are interested in is θ.
modulus and argument of z
The value of θ is π−α. Looking at the triangle with vertices at z, −2 and 0, we see that tanα=12. So θ=π−arctan12=2.6779… radians. (Note that arctan is often written as tan−1.)
A complex number z in polar form is given as r(cosθ+isinθ) and is often abbreviated as rcisθ, where r equals the modulus of the complex number. The value θ is called the argument of z, denoted by arg(z).
Note that r(cos(θ+2kπ)+isin(θ+2kπ)) represents the same complex number for every integer k. To achieve uniqueness, it is customary to restrict the argument to be nonnegative and less than 2π.
For the z given above, we can write z=5–√cis2.6779 if we round the argument of z to 4 decimal places.
To find the polar form of w, first note that | w |=12+22−−−−−−√=5–√ and the angle from the positive real axis to the line segment between 0 and w is arctan2=1.1071487.... Hence, w=5–√cis1.1071 if we round the argument of w to 4 decimal places.
In practice, conversion between rectangular form and polar form can only be done approximately (though the accuracy of approximation can be arbitrary) unless the arguments are values of angles whose exact values for sine and cosine are known; e.g. π6, π4, π3, etc.
Multiplying complex numbers in polar form
Let z=r1cisθ1 and w=r2cisθ2 be complex numbers in polar form. Then
zw=r1r2cis(θ1+θ2),
and if r2≠0,
zw=r1r2cis(θ1−θ2).
The first result can prove using the sum formula for cosine and sine. To prove the second result, rewrite zw as zw¯¯¯¯| w |2. The details are left as an exercise.