2) Simplify √-1-4/5-√-1

A.-1/24+19/24i
B.-21/26+1/26i
C.-19/24-1/24i
D.-1/26-21/26i

3) How many petals do the two rose curves r = 9 cos 61θ and r = 8 sin 60θ have combined?
A. 182
B. 242
C. 181
D. 131

4)What is true of the curve r = ƒ(θ) if its shape is unaffected when r is replaced by –r and θ is replaced by p – θ?
A. It's symmetric with respect to the x-axis.
B. It's symmetric with respect to the origin.
C. It's symmetric with respect to the line y = x.
D. It's symmetric with respect to the y-axis.

5) Simplify (√2-3i)^2
A.-5-6√2i
B.-7-6√2i
C. 2 − 9i
D.2 + 9i

Change 8 cis 240° to rectangular form.
A.-4-4√3i
B.-4√3-4i
C.-8√3-8i
D.-8-8√3i

√-1-4/5-√-1

(-4+i)/(5-i)

(-4+i)(5+i)
---------------
(5-i)(5+i)

(-21+i)/26

so, (B)
***********************
#3 - visit wolframalpha.com and see what the graphs look like if you have trouble algebraically.
***********************
r ==> -r reflects through the origin
θ ==> π-θ rotates 1/2 turn about the origin
Looks like (B) to me
***************************
(√2-3i)^2
2-6√2i+9i^2
....
*****************************
cis240° = -cos60°-i sin60°
...

2) To simplify √-1 - 4/5 - √-1, we can break it down step by step.

First, simplify √-1 by understanding that the square root of -1 is denoted by the imaginary unit "i". So, √-1 = i.

Next, simplify the expression √-1 - √-1 by substituting i for both square roots. i - i = 0.

Finally, subtract 4/5 from the result. 0 - 4/5 = -4/5.

Therefore, the simplified expression is -4/5. So, the correct answer is not listed among the options.

3) To find the combined number of petals for the two rose curves r = 9 cos 61θ and r = 8 sin 60θ, we first need to determine the number of petals for each individual curve.

For the curve r = 9 cos 61θ, the number of petals can be found by considering that the coefficient of θ in the polar equation represents the number of petals. In this case, the coefficient is 61. So, the number of petals for the first curve is 61.

Similarly, for the curve r = 8 sin 60θ, the coefficient of θ is 60. Thus, the number of petals for the second curve is 60.

To find the combined number of petals, we add the number of petals for each curve. 61 + 60 = 121.

Therefore, the correct answer is not listed among the options. It should be 121 petals instead.

4) If the shape of a curve r = ƒ(θ) remains unaffected when r is replaced by -r and θ is replaced by π - θ, it means that the resulting curve is symmetric with respect to the y-axis.

In this case, replacing r with -r reflects the curve across the pole (origin) and replacing θ with π - θ reflects the curve across the y-axis.

Since the curve's shape is unaffected, it remains symmetric about the y-axis.

So, the correct answer is (D) It's symmetric with respect to the y-axis.

5) To simplify (√2 - 3i)^2, we can use the formula (a - b)^2 = a^2 - 2ab + b^2.

Here, a = √2 and b = 3i.

First, square √2: (√2)^2 = 2.

Next, multiply 2 by -2ab: 2(-2)(√2)(3i) = -12√2i.

Finally, square 3i: (3i)^2 = 9i^2 = 9(-1) = -9.

Combine the results: 2 - 12√2i - 9 = -7 - 12√2i.

So, the correct answer is (B) -7 - 12√2i.

To change 8 cis 240° to rectangular form, we can convert it using the formula:

r cis θ = r * cos θ + r * sin θi

Here, r = 8 and θ = 240°, but we need to convert the angle to radians. Remember that π radians is equivalent to 180°.

To convert 240° to radians, we divide by 180° and multiply by π:

240° * (π/180°) = 4π/3

Now, substitute the values into the formula:

8 * cos (4π/3) + 8 * sin (4π/3)i

To simplify this expression, use the unit circle values for cosine and sine:

cos (4π/3) = -1/2
sin (4π/3) = -√3/2

Substitute the values back in:

8 * (-1/2) + 8 * (-√3/2)i

Simplify:

-4 - 4√3i

So, the correct answer is (A) -4 - 4√3i.