A complex number z1 has a magnitude ∣z1∣=24 and an angle θ1=5∘.Express z1 in rectangular form, as z1=a+bi.Round a and b to the nearest thousandth.z1=□+□i
Q. A complex number z1 has a magnitude ∣z1∣=24 and an angle θ1=5∘.Express z1 in rectangular form, as z1=a+bi.Round a and b to the nearest thousandth.z1=□+□i
Use Trigonometric Equations: To convert a complex number from polar to rectangular form, we use the equations a=∣z∣⋅cos(θ) and b=∣z∣⋅sin(θ), where a is the real part and b is the imaginary part of the complex number.
Convert Angle to Radians: Given ∣z1∣=24 and θ1=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.θ1 in radians = 5×(π/180)
Calculate Real Part a: Now we calculate the real part a:a=∣z1∣⋅cos(θ1)=24⋅cos(5⋅(π/180))
Round Real Part a: Calculate the value of a using a calculator:a≈24×cos(5×(π/180))≈24×0.99619≈23.90856Round a to the nearest thousandth:a≈23.909
Calculate Imaginary Part b: Now we calculate the imaginary part b:b=∣z1∣⋅sin(θ1)=24⋅sin(5⋅(π/180))
Round Imaginary Part b: Calculate the value of b using a calculator:b≈24×sin(5×(π/180))≈24×0.08716≈2.09184Round b to the nearest thousandth:b≈2.092
Write in Rectangular Form: Now we can write z1 in rectangular form:z1=a+bi=23.909+2.092i
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