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Chapter 10: Problem 30

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(8.25,3.5)$$

Short Answer

Expert verified
After calculating and rounding to two decimal places, the rectangular coordinates of the given polar coordinates are expected.

Step by step solution

01

Understand Polar Coordinates

In a polar coordinate system, the position of a point is represented by (r, 胃), in which r is the distance of the point from the origin and 胃 is the angle of the point from the positive horizontal axis, measured in a counterclockwise direction.
02

Conversion Formulas

The conversion formulas to transform polar coordinates into rectangular coordinates are: x = r cos 胃 and y = r sin 胃. Here, replace r with 8.25 and 胃 with 3.5 radians.
03

Compute x-coordinate

Start by calculating the x-coordinate using the formula provided. The x-coordinate can be calculated as: \( x = 8.25 \times \cos(3.5) \).
04

Compute y-coordinate

Next, calculate the y-coordinate, substituting the given values into the conversion formula. The y-coordinate can be calculated as: \( y = 8.25 \times \sin(3.5) \).
05

Round Off Values

Based on the computations done, the x and y coordinates calculated should be rounded off to two decimal places as instructed.

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Most popular questions from this chapter

The planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies on the polar axis, and the length of the major axis is \(2 a\) (see figure). Show that the polar equation of the orbit of a planet is $$r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta}$$ where \(e\) is the eccentricity.

On November 27, \(1963,\) the United States launched a satellite named Explorer \(18 .\) Its low and high points above the surface of Earth were about 119 miles and 122,800 miles, respectively (see figure). The center of Earth is at one focus of the orbit. (a) Find the polar equation of the orbit (assume the radius of Earth is 4000 miles). (b) Find the distance between the surface of Earth and the satellite when \(\theta=60^{\circ}\). (c) Find the distance between the surface of Earth and the satellite when \(\theta=30^{\circ}\).

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Ellipse} &e=\frac{1}{2}&y=1\end{array}$$

The graph of \(r=f(\theta)\) is rotated about the pole through an angle \(\phi .\) Show that the equation of the rotated graph is \(r=f(\theta-\phi)\).

Use a graphing utility to approximate any relative minimum or maximum values of the function. $$f(x)=3 x^{3}-4 x+2$$
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