I am having difficulty with two problems:
1) Evaluate (1+i)^12 by using DeMoivre's Theorem. Express the result in rectangular form.
So far I have: r=sqrt(1)^2+(1)^2 which simplified is sqrt2. Don't know how to proceed.
2) Write the polar equation theta = 45 degrees in rectangular form.
I don't even know how to proceed to do this.
Any help would be great!
(1+i)^12 is sqrt2 @ 45 to the 12th power, or sqrt2 ^12 @45x12
1@45 is .707i + .707j where i,j are unit vectors in the x,y directions.
x = r cos(45) ...(1)
y = r sin(45) ...(2)
Dividing (2) by (1):
y / x = tan(45) = 1
y = x.
find the distance between points at (6,-3) and (-1,4). The answer I got was 9.9 and it was wrong.
please help!
Sure, I can help you with both of these problems.
1) To evaluate (1 + i)^12 using DeMoivre's Theorem, we can start by expressing 1 + i in polar form. Remember that the polar form of a complex number is given by r(cos θ + i sin θ), where r is the magnitude of the complex number and θ is the angle it makes with the positive real axis.
To find the magnitude of 1 + i, we can use the Pythagorean theorem. The magnitude is given by √(1^2 + 1^2) = √2. So, 1 + i can be written as √2(cos θ + i sin θ), where θ is the angle it makes with the positive real axis.
Now, we can apply DeMoivre's Theorem: (r(cos θ + i sin θ))^n = r^n(cos nθ + i sin nθ). In this case, r = √2 and n = 12.
Using DeMoivre's Theorem, we have (√2)^12(cos (12θ) + i sin (12θ)). Now we need to find the value of θ.
Since 1 + i lies in the first quadrant, its angle θ is in the range [0°, 90°]. We can find θ by taking the inverse tangent of the imaginary part divided by the real part: θ = atan(1/1) = atan(1) = 45°.
Now, we can substitute θ = 45° into the expression: (√2)^12(cos (12×45°) + i sin (12×45°)).
Simplifying further, we have (√2)^12(cos 540° + i sin 540°) = (√2)^12((-1) + i × 0) = (-2)^6 + 0i = 64 + 0i = 64.
So, the result is 64 in rectangular form.
2) To write the polar equation θ = 45° in rectangular form, we need to convert it to the form x + yi = a + bi, where x and y are the real and imaginary parts respectively.
In polar form, θ = 45°, which means the angle is 45° and the magnitude (or distance from the origin) is not specified. Assuming the magnitude to be r, we can write the polar form as r(cos 45° + i sin 45°).
To convert this to rectangular form, we can use the identities cos α = x/r and sin α = y/r, where α is the angle and x, y, and r are as defined earlier.
In this case, x = r cos 45° and y = r sin 45°. Since the angle is 45°, cos 45° = sin 45° = 1/√2.
Substituting these values, we get x = r/√2 and y = r/√2. Rearranging the equations, we have r = x√2, r = y√2.
Hence, the rectangular form of the polar equation θ = 45° is x = y.
I hope this helps! Let me know if you have any further questions.