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A complex number z_(1) has a magnitude |z_(1)|=18 and an angle theta_(1)=122^(@). 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∣=18 \left|z_{1}\right|=18 and an angle θ1=122∘ \theta_{1}=122^{\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

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Q. A complex number z1 z_{1} has a magnitude ∣z1∣=18 \left|z_{1}\right|=18 and an angle θ1=122∘ \theta_{1}=122^{\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. Formulas Used: To convert from polar to rectangular form, we use the formulas a=r⋅cos⁡(θ)a = r \cdot \cos(\theta) and b=r⋅sin⁡(θ)b = r \cdot \sin(\theta), where rr is the magnitude and θ\theta is the angle.
  2. Calculate Real Part: First, calculate the real part aa: a=18×cos⁡(122∘)a = 18 \times \cos(122^\circ). Using a calculator, we find a≈18×(−0.5299)≈−9.5382a \approx 18 \times (-0.5299) \approx -9.5382.
  3. Calculate Imaginary Part: Now, calculate the imaginary part bb: b=18×sin⁡(122∘)b = 18 \times \sin(122^\circ). Using a calculator, we find b≈18×0.8480≈15.264b \approx 18 \times 0.8480 \approx 15.264.
  4. Round to Nearest Thousandth: Round aa and bb to the nearest thousandth.a≈−9.538a \approx -9.538 and b≈15.264b \approx 15.264.

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