找出每个复数i的模数和自变量。练习1至2。
问题1。z = – 1 – i√3
解决方案:
We have,
z = -1 – i√3
We know that, z = r (cosθ + i sinθ)
Therefore,
r cosθ = -1 —(1)
r sinθ = -√3 —-(2)
On Squaring and adding (1) and (2), we obtain
r2 (cos 2θ + sin 2θ) = (-1)2 + (-√3)2
r2 = 1 + 3
r = √4
Since r has to positive, Therefore r = 2
Putting r = 2 on (1) and (2), we get
cosθ = -1 / 2 and sinθ = -√3 / 2
Therefore, θ = – 2π / 3 (Since cosθ and sinθ both are negative, therefore θ lies in third quadrant)
Hence, modulus and argument of z = -1 – i√3 are 2 and – 2π / 3 respectively.
问题2. z =-√3+ i
解决方案:
We have,
z = -√3 + i
We know that, z = r (cosθ + i sinθ)
Therefore,
r cosθ = -√3 —(1)
r sinθ = 1 —-(2)
On Squaring and adding (1) and (2), we obtain
r2 (cos 2θ + sin 2θ) = (-√3)2 + (1)2
r2 = 3 + 1 (Since, cos 2θ + sin 2θ = 1)
r2 = 3 + 1
r = √4
Since r has to positive, Therefore r = 2
Putting r = 2 on (1) and (2), we get
cosθ = -√3 / 2 and sinθ = 1 / 2
Therefore, θ = 5π / 6 (Since cosθ negative and sinθ positive, therefore θ lies on second quadrant)
Hence, modulus and argument of z = -√3 + i are 2 and 5π / 6 respectively.
将练习3中给出的每个复数以极形式转换为:
问题3. 1 –我
解决方案:
We have z = 1 – i,
Let r cosθ = 1 —(1) and,
r sinθ = -1 —(2)
On Squaring and adding (1) and (2) , we obtain
r2 ( cos 2θ + sin 2θ ) = (1)2 + (-1)2
r2 = 2
r = √2 ( Since r has to be positive )
Putting r = √2 on (1) and (2) , we get
cosθ = 1 / √2 and sinθ = -1 / √2
Therefore, θ = – π / 4 ( Since cosθ positive and sinθ negative, therefore θ is negative as it lies on fourth quadrant)
Hence , z in polar form: z = r cosθ + i r sinθ = √2 (cos (- π / 4) + i sin (- π / 4)).
问题4. -1 + i
解决方案:
We have z = -1 + i,
Let r cosθ = -1 —(1) and,
r sinθ = 1 —(2)
On Squaring and adding (1) and (2), we obtain
r2 (cos 2θ + sin 2θ) = (-1)2 + (1)2
r2 = 2
r = √2 (Since r has to be positive)
Putting r = √2 on (1) and (2), we get
cosθ = -1 / √2 and sinθ = 1 / √2
Therefore, θ = 3π / 4 (Since cosθ negative and sinθ positive, therefore θ is positive as it lies on second quadrant)
Hence, z in polar form: z = r cosθ + i r sinθ = √2 (cos (3π / 4) + i sin (3π / 4)).
问题5. -1 – i
解决方案:
We have z = -1 – i,
Let r cosθ = -1 —(1) and,
r sinθ = -1 —(2)
On Squaring and adding (1) and (2), we obtain
r2 (cos 2θ + sin 2θ) = (-1)2 + (-1)2
r2 = 2
r = √2 ( Since r has to be positive )
Putting r = √2 on (1) and (2) , we get
cosθ = -1 / √2 and sinθ = -1 / √2
Therefore, θ = -3π / 4 (Since cosθ negative and sinθ negative, therefore θ is negative as it lies on third quadrant)
Hence, z in polar form: z = r cosθ + i r sinθ = √2 (cos (-3π / 4) + i sin (-3π / 4)).
问题6. -3
解决方案:
We have z = -3,
Let r cosθ = -3 —(1) and,
r sinθ = 0 —(2)
On Squaring and adding (1) and (2), we obtain
r2 (cos 2θ + sin 2θ) = (-3)2 + (0)2
r2 = 9
r = 3 (Since r has to be positive)
Putting r = 3 on (1) and (2), we get
cosθ = -3 / 3 and sinθ = 0 / 3
Therefore, θ = π
Hence, z in polar form: z = r cosθ + i r sinθ = 3(cos (π) + i sin (π)).
问题7.√3+ i
解决方案:
We have z = √3 + i,
Let r cosθ = √3 —(1) and,
r sinθ = 1 —(2)
On Squaring and adding (1) and (2) , we obtain
r2 (cos 2θ + sin 2θ) = (√3)2 + (1)2
r2 = 4
r = 2 (Since r has to be positive )
Putting r = 2 on (1) and (2), we get
cosθ = √3 / 2 and sinθ = 1 / 2
Therefore, θ = π / 6 ( Since cosθ positive and sinθ positive, therefore θ is positive as it lies on first quadrant)
Hence, z in polar form: z = r cosθ + i r sinθ = 2 (cos (π / 6) + i sin (π / 6)).
问题8.我
解决方案:
We have z = i,
Let r cosθ = 0 —(1) and,
r sinθ = 1 —(2)
On Squaring and adding (1) and (2) , we obtain
r2 (cos 2θ + sin 2θ) = (0)2 + (1)2
r2 = 1
r = 1 (Since r has to be positive)
Putting r = 1 on (1) and (2), we get
cosθ = 0 / 1 and sinθ = 1 / 1
Therefore, θ = π / 2
Hence, z in polar form: z = r cosθ + i r sinθ = cos (π / 2) + i sin (π / 2)