Write the complex number z=-2+2i in trigonometric form.
we want r(cos T + i sin T)
r = sqrt ((-2)^2 +2^2) =sqrt 8
=2 sqrt2
-2 = r cos T = 2 sqrt 2 cos T
cos T =-sqrt 2 / 2
T = 3 pi/4
2 = R sin T = 2 sqrt 2 sin T
sin T = sqrt2 /2
T = 3 pi/4 all right
so
z =2 sqrt 2[cos .75 pi+isin .75 pi]
To convert the given complex number z = -2 + 2i into trigonometric form, we need to find its magnitude (r) and argument (θ).
Step 1: Find the magnitude (r):
The magnitude (r) of a complex number is found using the formula: r = √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number, respectively.
In this case, a = -2 and b = 2:
r = √((-2)^2 + 2^2)
= √(4 + 4)
= √8
= 2√2
Step 2: Find the argument (θ):
The argument (θ) of a complex number can be found by using the formula: θ = arctan(b/a).
In this case, a = -2 and b = 2:
θ = arctan(2/-2)
= arctan(-1)
= -π/4 (or -45 degrees)
Therefore, the complex number z = -2 + 2i in trigonometric form is:
z = 2√2 * (cos(-π/4) + i * sin(-π/4))
To write the complex number z = -2 + 2i in trigonometric form, we can apply Euler's formula, which states that e^(iθ) = cos(θ) + isin(θ).
Step 1: Find the magnitude or modulus of the complex number by using the formula: |z| = √(x^2 + y^2), where x = real part and y = imaginary part.
For z = -2 + 2i, the magnitude is |z| = √((-2)^2 + 2^2) = √(4 + 4) = √8 = 2√2.
Step 2: Find the argument or angle θ of the complex number by using the formula: θ = arctan(y/x), where x = real part and y = imaginary part.
For z = -2 + 2i, the argument is θ = arctan(2/(-2)) = arctan(-1) = -π/4 radians.
Step 3: Now, we can express the complex number z in trigonometric form as z = |z| * e^(iθ), where |z| is the modulus and θ is the argument.
For z = -2 + 2i, the trigonometric form is:
z = 2√2 * e^(-iπ/4)
Therefore, the complex number -2 + 2i can be written as 2√2 * e^(-iπ/4) in trigonometric form.