find the trigonometric form of the following number:
-7 + 4i
I keep getting the wrong answer for these questions.
mine was:
square root of 65 (cos330.255 degrees + sin330.255i degrees)
z = r(cos ß + i(sin ß))
where r = √(x^2+y^2)
cos ß = x/r and sin ß = y/r
your z = -7 + 4i
so r = √65, which you had correctly done
cos ß = -7/√65 and sin ß = 4/√65
the angle in standard position is 29.7┼
the quadrant where the cosine is negative and the sine is postitive is the second quadrant
so ß = 180 - 29.7 = 150.26º
so -7+4i = √65(cos 150.26 + i(sin 150.26)
To find the trigonometric form of a complex number, we can use the magnitude and argument of the number.
Given the complex number -7 + 4i, let's start by finding its magnitude (r) and argument (θ):
Magnitude (r):
The magnitude of a complex number is the distance from the origin (0, 0) to the point representing the complex number in the complex plane.
The magnitude (r) can be calculated using the formula: r = √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number, respectively.
For -7 + 4i, the real part (a) is -7 and the imaginary part (b) is 4.
So, r = √((-7)^2 + 4^2) = √(49 + 16) = √65.
Argument (θ):
The argument of a complex number is the angle (in radians or degrees) between the positive real axis and the line connecting the origin to the point representing the complex number in the complex plane.
The argument (θ) can be calculated using the formula: θ = arctan(b/a), where a and b are the real and imaginary parts of the complex number, respectively.
For -7 + 4i, a = -7 and b = 4.
So, θ = arctan(4/-7) ≈ -0.519 radians (-29.746 degrees).
Now that we have the magnitude (r) = √65 and argument (θ) ≈ -0.519 radians (-29.746 degrees), we can write the trigonometric form of the complex number:
-7 + 4i = √65 * cos(-0.519) + √65 * i * sin(-0.519)
Note that the angle is negative because the complex number lies in the third quadrant of the complex plane.