Hello! Let’s convert this in fraction form. Keep first fraction, change division sign to multiplication sign, and flip the last fraction in reverse. It should look like this. 10/1 * 3/5. When you do that, you get 30/5, which is equivalent to 6. The quotient is 6.
The given complex number is
z = 1 + cos(2θ) + i sin(2θ), for -1/2π < θ < 1/2π
Part (i)
Let V = the modulus of z.
Then
V² = [1 + cos(2θ)]² + sin²(2θ)
= 1 + 2 cos(2θ) + cos²2θ + sin²2θ
Because sin²x + cos²x = 1, therefore
V² = 2(1 + cos2θ)
Because cos(2x) = 2 cos²x - 1, therefore
V² = 2(1 + 2cos²θ - 1) = 4 cos²θ
Because -1/2π < θ < 1/2π,
V = 2 cosθ PROVEN
Part ii.
1/z = 1/[1 + cos2θ + i sin 2θ]
![\frac{1}{z} = \frac{(1+cos2\theta - i\, sin2\theta)}{(1 + cos 2\theta + i\, sin 2\theta)(1+cos2\theta - i \,sin2\theta)}\\ = \frac{1+cos2\theta - i \,sin 2\theta}{(1+cos2\theta)^{2} + sin^{2}2\theta}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bz%7D%20%3D%20%5Cfrac%7B%281%2Bcos2%5Ctheta%20-%20i%5C%2C%20sin2%5Ctheta%29%7D%7B%281%20%2B%20cos%202%5Ctheta%20%2B%20i%5C%2C%20sin%202%5Ctheta%29%281%2Bcos2%5Ctheta%20-%20i%20%5C%2Csin2%5Ctheta%29%7D%5C%5C%20%3D%20%5Cfrac%7B1%2Bcos2%5Ctheta%20-%20i%20%5C%2Csin%202%5Ctheta%7D%7B%281%2Bcos2%5Ctheta%29%5E%7B2%7D%20%2B%20sin%5E%7B2%7D2%5Ctheta%7D%20)
The denominator is
![(1+cos2\theta)^{2}+sin^{2}2\theta \\ = 1+2cos2\theta+cos^{2}2\theta+sin^{2}2\theta \\ =2cos2\theta+2 \\ = 2(1+cos2\theta)](https://tex.z-dn.net/?f=%281%2Bcos2%5Ctheta%29%5E%7B2%7D%2Bsin%5E%7B2%7D2%5Ctheta%20%5C%5C%20%3D%201%2B2cos2%5Ctheta%2Bcos%5E%7B2%7D2%5Ctheta%2Bsin%5E%7B2%7D2%5Ctheta%20%5C%5C%20%3D2cos2%5Ctheta%2B2%20%5C%5C%20%3D%202%281%2Bcos2%5Ctheta%29)
Therefore
![\frac{1}{z} = \frac{1}{2} -i \frac{sin2\theta}{2(1+cos2\theta)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bz%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20-i%20%5Cfrac%7Bsin2%5Ctheta%7D%7B2%281%2Bcos2%5Ctheta%29%7D%20)
The real part of 1/ = 1/ (constant).
Answer: 3rd choice
To get from ABC to A'B'C' we reflected in the y axis.
To get from A'B'C' to A''B''C'' we translated down.
Yes I think it is correct
1. Pick two points on the line and determine their coordinates.
2. Determine the difference in y-coordinates of these two points (rise).
3. Determine the difference in x-coordinates for these two points (run).
4. Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).