9231 · 1.5
Polar coordinates flashcards
Revision flashcards for Cambridge 9231 Polar coordinates (syllabus 1.5). Flip, recall, then mark a real past-paper question.
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What are the polar coordinates $(r, \theta)$?
$r$ is the radial distance from the origin (the pole). $\theta$ is the angle measured anti-clockwise from the initial line (the positive x-axis).
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How do you convert from polar $(r, \theta)$ to Cartesian $(x, y)$ coordinates?
Use the formulae $x = r \cos \theta$ and $y = r \sin \theta$.
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How do you convert from Cartesian $(x, y)$ to polar $(r, \theta)$ coordinates?
Use the formulae $r^2 = x^2 + y^2$ (so $r = \sqrt{x^2+y^2}$) and $\tan \theta = \frac{y}{x}$. Be careful to place $\theta$ in the correct quadrant.
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What does a negative value of $r$ mean in a polar plot?
If $r$ is negative for a given $\theta$, you plot the point at a distance of $|r|$ from the pole, but in the opposite direction (i.e., along the line $\theta + \pi$).
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What is the general polar equation for a circle of radius $a$ centred at the origin?
$r = a$. The angle $\theta$ can take any value, but the distance from the pole is always $a$.
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What is the general polar equation for a straight line passing through the origin?
$\theta = \alpha$, where $\alpha$ is a constant. The angle is fixed, but the distance $r$ can be any real number.
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What is the formula for the area of a sector enclosed by a polar curve $r=f(\theta)$?
$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$, where the sector is bounded by the lines $\theta=\alpha$ and $\theta=\beta$.
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How do you find the tangents at the pole?
Solve the equation $r=f(\theta)=0$. The solutions for $\theta$ give the angles of the lines that are tangent to the curve at the origin (pole).
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What is a cardioid?
A heart-shaped polar curve with an equation of the form $r = a(1 \pm \cos \theta)$ or $r = a(1 \pm \sin \theta)$. It passes through the pole.
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What is a common trap when finding the area of a symmetrical polar curve?
Calculating the area for a smaller range of $\theta$ and then multiplying by a factor of symmetry. Forgetting to adjust the integration limits accordingly is a frequent error. For example, for a cardioid, integrate from $0$ to $\pi$ and double the result.
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What is the polar equation for a horizontal line $y=c$?
Since $y = r \sin \theta$, the equation is $r \sin \theta = c$, or $r = c \csc \theta$.
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What is the polar equation for a vertical line $x=c$?
Since $x = r \cos \theta$, the equation is $r \cos \theta = c$, or $r = c \sec \theta$.