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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
Formula to convert polar to Cartesian coordinates (x)?
$x = r \cdot cos(\theta)$
Formula to convert polar to Cartesian coordinates (y)?
$y = r \cdot sin(\theta)$
Formula to convert Cartesian to polar coordinates (r)?
$r = \sqrt{x^2 + y^2}$
Formula to convert Cartesian to polar coordinates (θ)?
$\theta = tan^{-1}(y/x)$ (Adjust quadrant!)
Polar form of a complex number?
$r(cos \theta + i sin \theta)$
How to find 'r' from a complex number a+bi?
$r = \sqrt{a^2 + b^2}$
How to find 'θ' from a complex number a+bi?
$\theta = tan^{-1}(b/a)$ (Adjust quadrant!)
How to represent the same point with a negative 'r'?
$(r, \theta) = (-r, \theta + \pi)$
How to represent the same point with coterminal angles?
$(r, \theta) = (r, \theta + 2\pi k)$
What are the equivalent cartesian coordinates when r = 2 and θ = 7π/6?
$x = 2cos(7π/6) = -√3$, $y = 2sin(7π/6) = -1$