Polar Coordinates
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| polar coordinates/ˈpəʊlə kəʊˈɔːdɪnəts/ | tọa độ cực |
| radians/ˈreɪdɪənz/ | radian |
| polar grid/ˈpəʊlə ɡrɪd/ | lưới tọa độ cực |
A different way to say "where"
- Usually we locate a point by "go right $x$, go up $y$" — rectangular coordinates.
- But a radar screen or a compass thinks differently: "how far, and in which direction?"
- That is the polar way: a distance and an angle.
- For anything that spins or radiates, polar coordinates are far more natural.
A point as (r, θ)
- Polar coordinates 极坐标 name a point by $(r, \theta)$.
- $r$ is the distance from the origin (the pole); $\theta$ is the angle from the positive x-axis.
- The angle is measured in radians 弧度 (or degrees), counter-clockwise.
- So $(2, \tfrac{\pi}{4})$ means "go out $2$ units at a $45°$ direction".
In polar coordinates $(r, \theta)$, the two numbers give…
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Polar coordinates use $r$ (distance from the origin) and $\theta$ (angle), unlike the $(x, y)$ of rectangular coordinates.
The polar grid
- A polar grid 极坐标网格 is a set of concentric circles (constant $r$) and radial lines (constant $\theta$).
- Reading a point is easy: find its circle, then its radial line.
- The origin is $r = 0$, the single centre of all the circles.
- This grid replaces the familiar square graph paper of rectangular coordinates.
Locate a point by radius and angle
Set the angle and the radius to place a point. Polar coordinates (r, theta) describe location by direction and distance.
To convert $(r, \theta)$ to rectangular $(x, y)$, use…
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The unit-circle relationship scales up by $r$: $x = r\cos\theta$ and $y = r\sin\theta$.
A point has $r = 2$ and $\theta = 0$. What is its $x$-coordinate?
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$x = r\cos\theta = 2\cos 0 = 2 \cdot 1 = 2$; and $y = 2\sin 0 = 0$.
Select all true statements about polar coordinates.
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Polar forms are not unique — many $(r, \theta)$ name the same point. The other three are correct.
Converting between systems
- Polar to rectangular: $x = r\cos\theta$ and $y = r\sin\theta$.
- Rectangular to polar: $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan\!\tfrac{y}{x}$ (mind the quadrant).
- These come straight from the unit-circle definitions, scaled by $r$.
- Switch to whichever system makes the problem simpler.
Every point has exactly one set of polar coordinates.
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False. Adding $2\pi$ to the angle (or using a negative $r$ with $\theta + \pi$) lands on the same point, so polar coordinates are not unique.
Not a unique address
- Unlike $(x, y)$, a polar point has many names.
- Adding $2\pi$ to $\theta$ lands on the very same point.
- A negative $r$ points in the opposite direction, giving yet another name.
- So $(2, 0)$, $(2, 2\pi)$, and $(-2, \pi)$ are all the same point.
Polar coordinates are not unique: one point has infinitely many $(r, \theta)$ names. When converting or solving, watch the quadrant and remember that $\arctan$ alone can land you in the wrong half of the plane.
Convert the polar point $(4, \tfrac{\pi}{2})$ to rectangular coordinates.
- $x = r\cos\theta = 4\cos\tfrac{\pi}{2} = 4 \cdot 0 = 0$.
- $y = r\sin\theta = 4\sin\tfrac{\pi}{2} = 4 \cdot 1 = 4$.
- So the point is $(0, 4)$ — straight up the y-axis.
Polar coordinates locate a point as $(r, \theta)$: a distance $r$ from the origin at angle $\theta$, plotted on a polar grid. Convert with $x = r\cos\theta$, $y = r\sin\theta$. Polar names are not unique — adding a full turn gives the same point.
Write from an x-coordinate of 5
- A polar point has r equals 5 and theta equals 0.
- Find the Cartesian x-coordinate.
A point has r equals 5 and theta equals 0. What is its x-coordinate?
5 times 1 is 5.
What is the x-coordinate?
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On the positive x-axis the x-coordinate equals r. Other options — 2 (the lesson x-coordinate); 0 (the angle used as a coordinate).