Geometry Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 10

1. To find the Cartesian x-value of a point expressed in polar coordinates, we multiply the radius by the cosine of the angle. In this case (Fig. 10-18), the radius equals 4 units, and the angle equals 5π/4 rad. We can recognize that angle as 225º. When we use a calculator to find the cosine of the angle and then multiply the result by 4, we get

4 cos 225º = -2.8284

accurate to four decimal places. Obviously, choices A and B are wrong! If we look at choice D and remember that π approximately equals 3.14159, then we see that -π/2 equals roughly -1.5708. That's wrong as well. Only choice C remains. If we take the square root of 8 (or the 1/2 power) with our calculator, round it off to four decimal places, and then multiply by -1, we get

-81/2 = -2.8284

By a process of elimination, we've determined that C has to be right! (For extra credit, can you prove that cos 225º equals exactly the negative of the square root of 8? Here's a hint: The Pythagorean theorem plays a role in the proof.)

2. To find the Cartesian y-value of a point in polar coordinates, we multiply the radius by the sine of the angle. In Fig. 10-18, once again, the radius is 4 units and the angle is 5π/4 rad or 225º. When we use a calculator to find the sine of the angle and then multiply by 4, we get

4 sin 225º = -2.8284

accurate to four decimal places. We can see straightaway that the correct choice is C, exactly the same value as we got for the x-coordinate. (For extra credit, can you prove that sin 225º equals exactly the negative of the square root of 8? Here's a hint: This proof follows the same "logic path" as the proof for the cosine does.)

3. First of all, I must make a correction to the wording of the question here! It should ask for the equation of the open-ended ray QP, not the open-ended ray QR, because the drawing doesn't show any point called R. Now that we've cleared up that error, we can see that the open-ended ray QP runs straight out from the origin in the direction for which θ = 5π/4 rad. The equation of the ray is therefore

θ = 5π/4

It's that simple, once we get our point names right! The answer is A.

4. Once again, I named a point wrong. Instead of point R, I meant to specify point P. If we construct a straight line passing through the origin (which we call point Q) and point P as shown in Fig. 10-18, we get a line through the origin (0,0) with a slope equal to 1. We know that the Cartesian xy-plane slope equals 1 because, for every unit we move in the positive x direction, we must go one unit in the positive y direction to stay on the line. (If you forgot the definition of slope, refer back to page 124 in Chapter 6.) We know that the y-intercept equals 0, because the line goes through the origin point (0,0). Therefore, in slope-intercept form as defined on page 124, the line has the equation

y = 1x + 0

which simplifies to

y = x

The correct choice is D.

5. A relation is a true mathematical function if and only if we can never find more than one value of the ordinate (dependent variable's coordinate) for any specific value of the abscissa (independent variable's coordinate). In the Cartesian xy-plane, we usually treat x as the independent variable (defined by the horizontal axis) and y as the dependent variable (defined by the vertical axis). For polar coordinates, this question asks us to treat the angle θ as the independent variable and the radius r as the dependent variable. When we examine each of the figures described here one by one we find that:

  • A straight vertical line through the origin doesn't represent a true function in either Cartesian coordinates or mathematician's polar coordinates (MPC), so choice A is wrong.
  • A straight horizontal line through the origin represents a true function in Cartesian coordinates, but not in MPC. Evidently, B is the right answer, but let's look at C and D to make sure.
  • A circle that doesn't contain the origin fails to represent a function in either Cartesian coordinates or MPC, so C is wrong.
  • A straight, horizontal line that doesn't pass through the origin represents a true function in both Cartesian coordinates and MPC. The question asks for something that works as a function only in the Cartesian system, not in MPC, so D is wrong.

Choice B is right after all!

6. In the situation of Fig. 10-19, the distance from point P to the origin (point Q) equals the length of the hypotenuse of a right triangle whose other two sides measure 6 units and 8 units. We can therefore use the Pythagorean theorem to calculate the distance from P to Q as

PQ = (62 + 82)1/2
= (36 + 64)1/2
= 1001/2
= 10 units

The measure of the angle θ doesn't matter in this case. The answer is B.
7. The distance PQ between P and the origin doesn't depend on the angle θ, but only on the radius r and the altitude h. Those two values haven't changed from the situation described in Question 6, so the distance PQ still equals 10 units. The correct choice is C.

8. An astronomer defines one minute of right ascension as 1/60 of an hour. One hour of right ascension equals 1/24 of a full circle around the heavens. Therefore, one minute of right ascension equals 1/24 of 1/60, or 1/1440, of a full circle. The answer is A.
9. In the situation shown by Fig. 10-20, the shaded plane contains the celestial equator. The angle θ represents celestial latitude. Direction V defines celestial latitude θ = 90º, going toward the north celestial pole. The correct choice is D.
10. In Fig. 10-20, the shaded plane W lies perpendicular to a line that runs toward the north and south celestial poles. That's the plane defined by the celestial equator, which coincides with the plane containing the earth's equator. The answer is A.