what is the trigonometric form of z=1+i?
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z = 1+i
r = √(1+1) = √2
tanØ = 1/1 = 1
Ø = 45°
1+i = √2(cos 45° + i sin45°)
what is z= -2-2square root of 3i in trigonometric form.
z= -2 - 2√3
r = √(4 + 12) = √16 or 4
tan Ø = -2√3/-2 , Ø must be in III
Ø = 240°
-2 - 2√3 i = 4(cos 240° + i sin 240°)
checking:
4(cos 240° + i sin 240°)
= 4(-.5 - √3/2)
= -2 - 2√3
To find the trigonometric form of a complex number, we can use the following steps:
Step 1: Calculate the modulus (magnitude) of the complex number:
The modulus of a complex number z = a + bi is given by |z| = √(a^2 + b^2).
In this case, the complex number z = 1 + i. So, the real part (a) is 1 and the imaginary part (b) is 1. Therefore, we calculate the modulus as follows:
|z| = √(1^2 + 1^2) = √2.
Step 2: Calculate the argument (angle) of the complex number:
The argument of a complex number z = a + bi can be found using the formula arctan(b/a) if a > 0, or arctan(b/a) + π if a < 0.
In our case, a = 1 and b = 1. Since a is positive, we can use the simple arctan(b/a) formula:
arg(z) = arctan(1/1) = arctan(1) = π/4.
Step 3: Express the complex number in the trigonometric form:
The trigonometric form of a complex number z = a + bi is given by z = |z| * (cos(arg(z)) + i * sin(arg(z))).
Using the values we calculated, the trigonometric form of z = 1 + i is:
z = √2 * (cos(π/4) + i * sin(π/4)).
So, the trigonometric form of z = 1 + i is √2 * (cos(π/4) + i * sin(π/4)).