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What are the differences between Cartesian and polar coordinates?
Cartesian: (x, y), uses perpendicular axes | Polar: (r, θ), uses distance and angle from origin
What are the differences between representing a point with positive and negative 'r' in polar coordinates?
Positive r: Point is in the direction of θ | Negative r: Point is in the opposite direction of θ
What are the differences between the real and imaginary axes in the complex plane?
Real axis: Represents the real part of a complex number | Imaginary axis: Represents the imaginary part of a complex number
What are the differences between rectangular and polar forms of a complex number?
Rectangular: a + bi | Polar: r(cos θ + i sin θ)
What are the differences between magnitude and argument of a complex number?
Magnitude: Distance from origin | Argument: Angle from positive real axis
What are the differences between degree and radian measure?
Degree: Unit of angle, 360° in a circle | Radian: Unit of angle, 2π radians in a circle
What are the differences between x and r*cos(θ)?
x: Cartesian coordinate | r*cos(θ): Equivalent in polar coordinates
What are the differences between y and r*sin(θ)?
y: Cartesian coordinate | r*sin(θ): Equivalent in polar coordinates
What are the differences between tan⁻¹(y/x) and the actual angle θ?
tan⁻¹(y/x): Gives angle in quadrants I or IV | θ: Requires quadrant adjustment for correct angle
What are the differences between a complex number and a real number?
Complex number: Has real and imaginary parts (a + bi) | Real number: Only has a real part
Convert (3, π/6) from polar to Cartesian coordinates.
x = 3*cos(π/6) = 3√3/2, y = 3*sin(π/6) = 3/2
Convert (-1, 1) from Cartesian to polar coordinates.
r = √( (-1)² + 1²) = √2, θ = tan⁻¹(1/-1) = -π/4 + π = 3π/4
Find an equivalent polar coordinate representation for (2, π/2) with a negative 'r'.
(-2, π/2 + π) = (-2, 3π/2)
Convert the complex number 1 + i to polar form.
r = √(1² + 1²) = √2, θ = tan⁻¹(1/1) = π/4. Polar form: √2(cos(π/4) + i sin(π/4))
Convert the polar coordinate (4, 5π/3) to cartesian coordinates.
x = 4cos(5π/3) = 2, y = 4sin(5π/3) = -2
Convert the cartesian coordinates (0, -5) to polar coordinates.
r = √(0² + (-5)²) = 5, θ = 3π/2
Represent the polar coordinate (6, π/4) with an angle between 2π and 4π.
(6, π/4 + 2π) = (6, 9π/4)
If z = 3(cos(π/6) + i sin(π/6)), find z in the form a + bi.
z = 3(√3/2 + i(1/2)) = (3√3/2) + (3/2)i
If z = -2 + 2i, find z² in the form a + bi.
z² = (-2 + 2i)² = 4 - 8i - 4 = -8i
Express the complex number z = 5(cos(π) + i sin(π)) in cartesian form.
z = 5(-1 + i(0)) = -5
Why can a single point have multiple polar coordinate representations?
Due to the periodic nature of angles and the possibility of negative radial distances.
Explain how changing the sign of 'r' affects the point's location.
Changing the sign of 'r' reflects the point across the origin (pole).
Why is quadrant adjustment important when converting from Cartesian to polar coordinates?
The arctangent function only gives angles in the 1st and 4th quadrants; adjustment ensures the correct quadrant for θ.
Explain the relationship between complex numbers and the complex plane.
A complex number a + bi is represented by the point (a, b) on the complex plane.
How are complex numbers related to polar coordinates?
The real and imaginary parts of a complex number can be expressed using polar coordinates: x = r*cos(θ), y = r*sin(θ).
When is it useful to use polar coordinates?
Polar coordinates are useful for circular or rotational symmetry.
Why is it important to check if angles are in degrees or radians?
Using the wrong units can lead to incorrect conversions and calculations.
What are the high-priority topics for exams?
Converting between polar and Cartesian coordinates, understanding multiple representations of polar coordinates, complex numbers in polar form, and quadrant awareness.
What is the standard form of a complex number?
a + bi
What is the polar representation of a complex number?
r(cos θ + i sin θ)