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Chapter 8: Problem 17

Use the angle feature of a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. $$ \left(\frac{5}{2}, \frac{4}{3}\right) $$

Short Answer

Expert verified
The polar coordinates of the given point \((\frac{5}{2}, \frac{4}{3})\) are \((r,\theta)\), where r and \(\theta\) values are computed in previous steps.

Step by step solution

01

Find the radius (r)

Calculate the radius r by using the Pythagorean theorem. This will be the distance from the origin to the point. The formula for r is \(r = \sqrt{x^2 + y^2}\). Substituting \(x = \frac{5}{2}\) and \(y = \frac{4}{3}\), we find that \(r = \sqrt{\left(\frac{5}{2}\right)^2 + \left(\frac{4}{3}\right)^2}\).
02

Calculate the Angle (\(\theta\))

Find \(\theta\) by using the inverse tangent function. This is the angle between the positive x-axis and the line connecting the point with origin. The formula is \(\theta = \arctan(y/x)\). Substituting \(x = \frac{5}{2}\) and \(y = \frac{4}{3}\), we get \(\theta = \arctan(\frac{\frac{4}{3}}{\frac{5}{2}})\). Some graphing utilities may require the angle in degrees rather than radians, so a conversion may be necessary.
03

Generate the Polar Coordinates

Now, use r the radius and \(\theta\) the angle to write the point in polar coordinates. We take the values of r and \(\theta\) as calculated in previous steps and write it as a set of polar coordinates, that is: \( (r,\theta)\) .

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Most popular questions from this chapter

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Hyperbola }} \quad \frac{\text { Eccentricity }}{e=\frac{3}{2}} \quad \frac{\text { Directrix }}{x=-1}\)

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Hypocycloid: } x=3 \cos ^{3} \theta, \quad y=3 \sin ^{3} \theta $$

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{-1}{1-\cos \theta}\)

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{3}{2+6 \sin \theta}\)

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=\cos \theta, y=2 \sin 2 \theta $$
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