The Modulus, Argument and Polar Form of a Complex Number
26 important questions on The Modulus, Argument and Polar Form of a Complex Number
How is the complex number \(z = a + bi\) represented?
What angle does the complex number \(z\) make with the positive direction of the Re-axis?
Using trigonometric identities, what does \(\cos\theta\) and \(\sin\theta\) represent in the complex number representation?
- \(\sin\theta = b / r\)
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How is the complex number \(z = a + bi\) represented using trigonometric functions?
What is the **polar form** of a complex number?
-
What are the **rules** for multiplying two complex numbers in polar form?
- Add the arguments
How can Equation (3) be simplified using **trigonometry**?
- z₁z₂ results from trigonometry identities
What are the **components** of the polar form of a complex number?
-
How is the **product of modulus** of two complex numbers calculated?
- Results in the modulus of the product of the complex numbers
What does the **angle** between the positive Re-axis and a complex number line represent?
- Denoted as arg(z)
What is the significance of the **modulus** in the **polar form** of a complex number?
- Denoted as r in the polar form
What is the rule for dividing two complex numbers in polar form?
- Subtract the argument of the denominator from the argument of the numerator
If 1 z = r1(cosθ1 + isinθ1) and 2 z = r2(cosθ2 + isinθ2), what is the formula for dividing these two complex numbers in polar form?
- (sinθ1 - θ2)(r1/r2)
What is Euler's formula for complex exponential functions?
What is the argument of the product of two complex numbers in polar form?
In Euler's formula, what is the relationship established between trigonometric functions and complex exponential functions?
How can you convert the complex number 2 + 2i into its polar form?
What is the significance of Euler's formula in complex analysis?
How do you change the complex number 3i - into its polar form?
How can we rewrite the polar form of a complex number into exponential form using Euler’s formula?
- Where θ = ar g z
How do we find the value of r in the exponential form using the modulus of both sides?
- Using trigonometric identities
- r = z
How can we consider the equation i*z = r*e^(i*θ) as a parametric representation of a circle?
- Exponential form as another way of polar form
What rules for multiplying and dividing complex numbers in polar form apply to the exponential form?
- Multiplying and dividing complex numbers
Express 1/3 * i = -1 - i in the form i*z = r*e^(i*θ).
- Find r and θ values
Express e^(5π*i/3) in the form z = a + bi.
- e^(5π*i/3) = -1/2 + sqrt(3)/2i
Prove cos(t) + i * sin(t) = e^(it).
- Using complex exponential
- Derive from Taylor series
- Cos(t) + i * sin(t) = e^(it)
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