put this in trigonometric form:
-3-i
Try writing it as the product of a scalar sqrt(10) and an expression of the form e^(i*theta), where theta is an angle in the third quadrant. It will be 180 + arctan 1/3 = 198.435 degrees or 3.4533 radians
sqrt(10)*exp(3.4533 i) = -3 - i
-3 + e[3/2)i pi]
or sqrt 10* exp (i A)
where A = pi + arctan (1/4)= 3.38657 radians
Or maybe they would like with sin A and cos A
where R e^iA = R cos A + i R sin A
then
-3 = R cos A
-1 = R sin A
so
R^2 sin^2 A + R^2 cos^2 A = 1+9 = 10
R^2 (1) = 10
R = sqrt 10 like he said
then cos A = -3/sqrt 10
and sin A = -1/sqrt 10
so
A is in quadrant 3 and is 198.43 degrees or 3.46 radians
so
-3 -i = sqrt 10 * [cos 3.46 + i sin 3.46]
Damian, I think that is what they wanted.
Here is a the same kind of question from earlier.
http://www.jiskha.com/display.cgi?id=1211758996
To express the complex number -3 - i in trigonometric form, we need to find its magnitude (or modulus) and argument (or angle).
1. Magnitude (Modulus):
The magnitude of a complex number is calculated using the formula:
|z| = sqrt( (real part)^2 + (imaginary part)^2 )
For -3 - i:
|z| = sqrt( (-3)^2 + (-1)^2 )
= sqrt(9 + 1)
= sqrt(10)
2. Argument (Angle):
The argument (θ) of a complex number can be found using the inverse tangent function:
θ = atan( (imaginary part) / (real part) )
For -3 - i:
θ = atan( (-1) / (-3) )
= atan(1/3)
≈ 18.435° (approximately)
Trigonometric Form:
In the trigonometric form, a complex number is expressed as:
z = |z| * (cosθ + i*sinθ)
For -3 - i:
z = sqrt(10) * (cos(18.435°) + i*sin(18.435°))
Therefore, the trigonometric form of -3 - i is ≈ sqrt(10) * (cos(18.435°) + i*sin(18.435°)).