Two sources of electrical alternating current (AC) have voltages of 100 V and 180 V respectively.?

Question

Two sources of electrical alternating current (AC) have voltages of 100 V and 180 V respectively. The first source has a phase shift of 45 degrees and the second phase shift of 135 degrees.

a) Find the voltage of the combined source? Answer: 206 Voltsb) Find the phase shift of the combined source? Answer: 105.9 degrees

2) Given z1= 4cis(-pi/3) and z2=3cis(5pi/6)Find the expressions for the following in a+bi form:a) z1*z2 Answer: 0+12ib) (z1/z2)^3 Answer: 0+64/27i

Can you show me how this is worked out?

Answer 1

phase referenced to what? I'll shift the first to 0º, so the second is at 90º.  Assuming those voltages are RMS, and you want RMS in the answer... (but they could be peak, answer would be the same)

first is 100√2sinωt

second is –180√2cosωt

the sum is 20√2(–9cosωt+5sinωt)

I'll use the formula

   Acosωt + Bsinωt = (√(A²+B²))cos(ωt–arctan(B/A))

sum = 20√2(√(5²+9²))cos(ωt–arctan(–5/9)

sum = (20√212)cos(ωt+29º)

so peak is 20√212, RMS is 20√212/√2 = 206 volts

phase is –29º + 90º(because cos) + 45º = 106º