1 POLAR COORDINATES Polar coordinate system: a pole (fixed point) and a polar axis (directed ray with endpoint at pole). rectangular coordinates ⇒ polar coordinates polar coordinates ⇒ rectangular coordinates = √ 2 + 2 , = = = The angle, θ, is measured from the polar axis to a line that passes through the point and the pole. If the angle is measured in a counterclockwise direction, the angle is positive. If the angle is measured in a clockwise direction, the angle is negative. The directed distance, r, is measured from the pole to point P. If point P is on the terminal side of angle θ, then the value of r is positive. If point P is on the opposite side of the pole, then the value of r is negative. Problem : P (x, y) = (1, 3). Express it in polar coordinates (r, θ) two different ways such that 0≤ θ < 2(r, θ) = (2, /3), (- 2, 4/3) . Problem : P(x, y) = (-4, 0). Express it in polar coordinates (r, θ) two different ways such that 0≤ θ < 2. (r, θ) = (4, ),(- 4, 0) . Problem : P (x, y) = (-7, -7), express it in polar coordinates (r, θ) two different ways such that 0≤ θ < 2. (r, θ) = (98, 5/4),(- 98, /4) . Problem : Given a point in polar coordinates (r, θ) = (3, /4), express it in rectangular coordinates (x, y) . (x, y) = (3√2/2, 3√2/2) . Problem : How many different ways can a point be expressed in polar coordinates such that r > 0 ? An infinite number. (r, θ) = (r, θ +2n) , where n is an integer. Problem : Transform the equation x 2 + y 2 + 5x = 0 to polar coordinate form. x 2 + y 2 + 5x = 0 r 2 + 5(r cos θ) = 0 r ( r + 5 cos θ) = 0 The equation r = 0 is the pole. Thus, keep only the other equation: r + 5 cos θ = 0 The location of a point can be named using many different pairs of polar coordinates. ← three different sets of polar coordinates for the point P (5, 60°). The distance r and the angle are both directed--meaning that they represent the distance and angle in a given direction. It is possible, therefore to have negative values for both r and . However, we typically avoid points with negative r , since they could just as easily be specified by adding to .
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POLAR COORDINATES - Uplift Education · 2015. 2. 23. · 1 POLAR COORDINATES Polar coordinate system: a pole (fixed point) and a polar axis (directed ray with endpoint at pole). rectangular
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POLAR COORDINATES
Polar coordinate system: a pole (fixed point) and a polar axis (directed ray with endpoint at pole).
The angle, θ, is measured from the polar axis to a line that passes through the point and the pole. If the angle is measured in a counterclockwise direction, the angle is positive. If the angle is measured in a clockwise direction, the angle is negative. The directed distance, r, is measured from the pole to point P. If point P is on the terminal side of angle θ, then the value of r is positive. If point P is on the opposite side of the pole, then the value of r is negative.
Problem : P (x, y) = (1, 3). Express it in polar coordinates (r, θ) two different ways such that 0≤ θ < 2
(r, θ) = (2, /3), (- 2, 4/3) .
Problem : P(x, y) = (-4, 0). Express it in polar coordinates (r, θ) two different ways such that 0≤ θ < 2.
(r, θ) = (4, ),(- 4, 0) .
Problem : P (x, y) = (-7, -7), express it in polar coordinates (r, θ) two different ways such that 0≤ θ < 2.
(r, θ) = (98, 5/4),(- 98, /4) .
Problem : Given a point in polar coordinates (r, θ) = (3, /4), express it in rectangular coordinates (x, y) . (x, y) = (3√2/2, 3√2/2) .
Problem : How many different ways can a point be expressed in polar coordinates such that r > 0 ?
An infinite number. (r, θ) = (r, θ +2n) , where n is an integer.
Problem : Transform the equation x2 + y2 + 5x = 0 to polar coordinate form.
x2 + y2 + 5x = 0 r2 + 5(r cos θ) = 0
r ( r + 5 cos θ) = 0 The equation r = 0 is the pole. Thus, keep only the other equation: r + 5 cos θ = 0
The location of a point can be named using many different pairs of polar coordinates. ← three different sets of polar coordinates for the point P (5, 60°).
The distance r and the angle 𝜃 are both directed--meaning that they represent the
distance and angle in a given direction. It is possible, therefore to have negative values for
both r and 𝜃. However, we typically avoid points with negative r , since they could just as
easily be specified by adding to 𝜃 .
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Problem : Transform the equation r = 4sin θ to Cartesian coordinate form. What is the graph? Describe it fully!!!