Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A complex number 
z_(1) has a magnitude 
|z_(1)|=24 and an angle 
theta_(1)=5^(@).
Express 
z_(1) in rectangular form, as 
z_(1)=a+bi.
Round 
a and 
b to the nearest thousandth.

z_(1)=◻+◻i

A complex number z1 z_{1} has a magnitude z1=24 \left|z_{1}\right|=24 and an angle θ1=5 \theta_{1}=5^{\circ} .\newlineExpress z1 z_{1} in rectangular form, as z1=a+bi z_{1}=a+b i .\newlineRound a a and b b to the nearest thousandth.\newlinez1=+i z_{1}=\square+\square i

Full solution

Q. A complex number z1 z_{1} has a magnitude z1=24 \left|z_{1}\right|=24 and an angle θ1=5 \theta_{1}=5^{\circ} .\newlineExpress z1 z_{1} in rectangular form, as z1=a+bi z_{1}=a+b i .\newlineRound a a and b b to the nearest thousandth.\newlinez1=+i z_{1}=\square+\square i
  1. Use Trigonometric Equations: To convert a complex number from polar to rectangular form, we use the equations a=zcos(θ)a = |z| \cdot \cos(\theta) and b=zsin(θ)b = |z| \cdot \sin(\theta), where aa is the real part and bb is the imaginary part of the complex number.
  2. Convert Angle to Radians: Given z1=24\lvert z_{1} \rvert = 24 and θ1=5\theta_{1} = 5 degrees, we first convert the angle to radians because the trigonometric functions in most calculators use radians. To convert degrees to radians, we multiply by π/180\pi/180.\newlineθ1\theta_{1} in radians = 5×(π/180)5 \times (\pi/180)
  3. Calculate Real Part aa: Now we calculate the real part aa:a=z1cos(θ1)=24cos(5(π/180))a = |z_{1}| \cdot \cos(\theta_{1}) = 24 \cdot \cos(5 \cdot (\pi/180))
  4. Round Real Part aa: Calculate the value of aa using a calculator:\newlinea24×cos(5×(π/180))24×0.9961923.90856a \approx 24 \times \cos(5 \times (\pi/180)) \approx 24 \times 0.99619 \approx 23.90856\newlineRound aa to the nearest thousandth:\newlinea23.909a \approx 23.909
  5. Calculate Imaginary Part bb: Now we calculate the imaginary part bb:b=z1sin(θ1)=24sin(5(π/180))b = |z_{1}| \cdot \sin(\theta_{1}) = 24 \cdot \sin(5 \cdot (\pi/180))
  6. Round Imaginary Part bb: Calculate the value of bb using a calculator:\newlineb24×sin(5×(π/180))24×0.087162.09184b \approx 24 \times \sin(5 \times (\pi/180)) \approx 24 \times 0.08716 \approx 2.09184\newlineRound bb to the nearest thousandth:\newlineb2.092b \approx 2.092
  7. Write in Rectangular Form: Now we can write z1z_{1} in rectangular form:\newlinez1=a+bi=23.909+2.092iz_{1} = a + bi = 23.909 + 2.092i

More problems from Write equations of cosine functions using properties