Trigonometry Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 7
1. One minute (1 m) of celestial right ascension equals 1/60 of an hour (1/60 h). In astronomy, an "hour" of right ascension represents 15 of arc going along the celestial equator, or 1/24 of a full circle around the heavens. Therefore, 1 m equals (1/24) x (1/60) = 1 / 1440 of a full circle. The correct choice is D.
2. Figure 7-13 shows a polar coordinate system with a vertical z axis added, running through the origin at a right angle to the polar plane containing the x and y axes. By definition, this arrangement constitutes a set of cylindrical coordinates for locating points in three-space. The correct choice is B.
3. If we construct a straight line segment from point P to the origin in Fig. 7-13, we get a right triangle whose base connects the origin to P', whose vertical side connects P' to P, and whose hypotenuse runs from the origin upward at a slant to P. If we call the base length a, the height b, and the hypotenuse c, then the Pythagorean theorem tells us that

c = (a2 + b2)1/2

The value of c equals the distance from the origin to P as measured along a straight line through space. We know that a = 18 units and b = 24 units. Therefore

c = (182 + 242)1/2
= (324 + 576)1/2
= 9001/2
= 30 units

The answer is B.

4. Let's find the xy-plane coordinates of P' first, as the first hint in the question suggests. In mathematician's polar coordinates (MPC), point P' corresponds to an ordered pair (θ,r) = (3π/4,18) in this particular case. The angle θ goes counterclockwise from the +x axis to point P' in the xy-plane. (It's shown as 3π/4 radians.) The radius r goes outward from the origin to P' in the xy-plane. (It's shown as 18 units.) We're told to consider these coordinate values exact. Let's recall the conversion formulas for changing a coordinate in MPC to a coordinate in the Cartesian xy-plane. We have

x = r cos θ

and

y = r sin θ

Setting π = 3.1416, we can use a calculator set for radians to derive the x-coordinate as

x = 18.00 cos (3π/4)
= 18.00 cos 2.3562
= 18.00 x (-0.7071)
= -12.73

rounded off to two decimal places. We derive the y-coordinate as

y = 18.00 sin (3π/4)
= 18.00 sin 2.3562
= 18.00 x 0.7071
= 12.73

again rounded off to two decimal places. We know that the z-coordinate equals 24.00, because that's the vertical distance from the xy-plane to point P as we proceed straight upward from P' parallel to the z axis. Therefore, the coordinates of P in Cartesian xyz-space are

P = (-12.73,12.73,24.00)

The answer is C.

5. We want to determine the measure of the angle between the line segment connecting the origin to P' (the base of the right triangle that we described when we solved Question 3) and the line segment connecting the origin to P (the hypotenuse of the same right triangle). We can accomplish this mission with the help of the knowledge that we gained in Chapter 4. We've been given a base length of 18 units and a height of 24 units. The measure of the angle θ in question is therefore

θ = Arctan (24/18)
= Arctan 1.333
= 53

rounded off to the nearest degree. The correct choice is D.

6. The magnitude of a vector in Cartesian coordinates is the same thing as the distance of its end point from the origin. We can use the Pythagorean distance formula that we learned earlier in this course to determine the magnitude of any vector in the Cartesian xy-plane. In this case, we have the vector

a = (5,-7)

so the x-coordinate equals 5 and the y-coordinate equals -7. When we input these numbers into the magnitude formula, we get

|a| = [52 + (-7)2]1/2
= (25 + 49)1/2
= 741/2
= 8.602

rounded off to three decimal places. The answer is A.

7. To calculate the MPC direction angle of the Cartesian xy-plane vector a = (5,-7), we should first note that x = 5 and y = -7. We can then take advantage the formula described in the chapter text, for points in which x > 0 and y < 0. If we call our MPC direction angle θ going counterclockwise from the +x axis, then we have

θ = 360 + Arctan (y/x)
= 360 + Arctan (-7/5)
= 360 + Arctan -1.400
= 360 + (-54)
= 360 - 54
= 306

The correct choice is A.

8. We have the vector a = (7π/6,5.00) in the MPC plane. We can call θ = 7π/6 and r = 5.00. Using the conversion formulas, we find that the Cartesian x-coordinate is

x = r cos θ
= 5.00 cos (7π/6)
= 5.00 x (-0.866)
= -4.33

and the Cartesian y-coordinate is

y = r sin θ
= 5.00 sin (7π/6)
= 5.00 x (-0.500)
= -2.50

The complete Cartesian xy-coordinate expression for our vector is

a = (-4.33,-2.50)

The correct choice is C.

9. The radius of any vector in navigator's polar coordinates (NPC) is identical to the MPC radius. Therefore, we know straightaway that r = 5.00. In order to figure out the NPC angle, we can use the formula given in Chapter 6 for converting MPC angles to NPC angles. In this case, we have the MPC angle θ = 7π/6. Before we use the conversion formula, we should multiply this angle by 180/π to get the equivalent in degrees. We obtain

θ = (7π/6) x (180/π)
= 7 x 180/6
= 7 x 30
= 210

Now we can use the conversion formula to get the angle α in NPC. Because our MPC angle is greater than 90 and less than 360, we should employ the formula

α = 450 - θ
= 450 - 210
= 240

We've determined that the NPC coordinates of our vector are

a = (240,5.00)

The answer is C.

10. We have the vector a in Cartesian xyz-space, where x = 5, y = -3, and z = -2. We can calculate the vector's magnitude as

|a| = (x2 + y2 + z2)1/2
= [52 + (-3)2 + (-2)2]1/2
= (25 + 9 + 4)1/2
= 381/2
= 6.16

rounded off to the nearest hundredth of a unit. The correct choice is D.