1.) Write the complex number in trigonometric form r(cos theta + i sin theta) with theta in the interval [0°, 360°).

9 sqrt 3 + 9i

2.) Find the product. Write the product in rectangular form, using exact values.
[4 cis 30°] [5 cis 120°]

3.) [4(cos 135° + i sin 135°)][6(cos 225° + i sin 225°)]

4.) Find the following quotient, and write the quotient in rectangular form, using exact values.
[25(cos 240° + i sin 240°)]/[5(cos 60° + i sin 60°)]

I need help setting this up please.

#1

You have
y = 9
x = 9√3
So, tanθ = 1/√3
r = 18
You should be able to recognize a 30-60-90 right triangle...

#2
(4*5)cis(30°+120°)

#3 is just the same.

#4 is the same, but divide r, and subtract angles.

Sure! I can help you set up these problems. Let's break them down one by one:

1.) To write the complex number 9√3 + 9i in trigonometric form, we need to find the magnitude (r) and the angle (theta) of the complex number.

- The magnitude (r) is found using the formula: r = sqrt(a^2 + b^2), where a and b are the real and imaginary parts respectively.
In this case, a = 9√3 and b = 9.
So, r = sqrt((9√3)^2 + 9^2) = sqrt(243 + 81) = sqrt(324) = 18.

- The angle (theta) is found using the formula: theta = arctan(b/a), where b and a are the imaginary and real parts respectively.
In this case, a = 9√3 and b = 9.
So, theta = arctan(9/9√3) = arctan(1/√3) = arctan(√3/3).

Now, we can write the complex number in trigonometric form:
9√3 + 9i = 18 * (cos(arctan(√3/3)) + i * sin(arctan(√3/3))).

2.) To find the product of [4 cis 30°] and [5 cis 120°], we can use the formula for multiplying complex numbers in trigonometric form: r1(cos(theta1) + i*sin(theta1)) * r2(cos(theta2) + i*sin(theta2)) = r1*r2 * (cos(theta1 + theta2) + i*sin(theta1 + theta2)).

In this case, r1 = 4, theta1 = 30°, r2 = 5, and theta2 = 120°.
So, the product would be: 4 * 5 * (cos(30° + 120°) + i * sin(30° + 120°)).

3.) To find the product of [4(cos 135° + i sin 135°)] and [6(cos 225° + i sin 225°)], we can use the same formula as in the previous question:

[4(cos 135° + i sin 135°)][6(cos 225° + i sin 225°)] = 4 * 6 * (cos(135° + 225°) + i * sin(135° + 225°)).

4.) To find the quotient of [25(cos 240° + i sin 240°)] divided by [5(cos 60° + i sin 60°)], we can again use the same formula:

[25(cos 240° + i sin 240°)]/[5(cos 60° + i sin 60°)] = 25/5 * (cos(240° - 60°) + i * sin(240° - 60°)).

Hope this helps! Let me know if you have any further questions.