Trigonometry Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 6
1. When we want to convert a point (θ,r) from mathematician's polar coordinates (MPC) to a point (x,y) in Cartesian coordinates, we use the formulas

x = r cos θ

and

y = r sin θ

In this case, our MPC point is (θ,r) = (5π/4,21/2). Therefore

x = 21/2 cos (5π/4)

As things work out, cos (5π/4) equals exactly -1/(21/2). (Check it out on your calculator leaving all the digits intact through the entire process, and you'll see this coincidence for yourself. Alternatively, if you want to play the role of the pure mathematician, you can use the Pythagorean theorem and the unit-circle paradigm to prove it.) Therefore

x = 21/2 x [-1/(21/2)]
= -1

We can also calculate

y = 21/2 sin (5π/4)

Again coincidentally, we find that sin (5π/4) = -1/(21/2). Therefore

y = 21/2 x [-1/(21/2)]
= -1

We can conclude that (x,y) = (-1,-1). The correct choice is C. As an "extra credit" exercise, try converting this Cartesian coordinate back to MPC to verify that the process works perfectly the other way as well.

2. When we want to convert a point (x,y) in the fourth quadrant of a Cartesian system (where our specified point happens to lie) to its counterpart (θ,r) in MPC, we use the formulas

θ = 2π + Arctan (y/x)

and

r = (x2 + y2)1/2

The question calls for an angle value in degrees. Mathematicians don't normally use degrees in their version of the polar coordinate system, but let's heed the instructions and modify the angle formula to

θ = 360 + Arctan (y/x)

Our point has the Cartesian coordinates (x,y) = (4,-3). Substituting y = -3 and x = 4 into the above equation and employing a calculator (set for degrees, not radians) to obtain the Arctangent, we get

θ = 360 + Arctan (-3/4)
= 360 + Arctan (-0.75)
= 360 + (-37)
= 323

When we calculate the radius using the Pythagorean formula for that coordinate, we get

r = [42 + (-3)2]1/2
= (16 + 9)1/2
= 251/2
= 5

We conclude that the MPC equivalent of the Cartesian coordinate (x,y) = (4,-3), using degrees for the angular measure, is

(θ,r) = (323,5)

The answer is D.

3. First, we should note that this question talks about the same point as Question 2 describes, so we already know that the MPC coordinates are

(θ,r) = (323,5)

When we want to convert an angle θ from the MPC system to an angle α in the navigator's polar coordinate (NPC) system, we use the formula

α = 450 - θ

when 90 < θ < 360, as we have here. Therefore

α = 450 - 323
= 127

The radius in NPC equals the radius in MPC; both quantities refer to the distance from the origin, so we still have r = 5. We've determined that the NPC equivalent of the Cartesian coordinate (x,y) = (4,-3) is

(α,r) = (127,5)

The correct choice is C.

4. In Fig. 6-10, line L runs through the origin in the angular direction 3π/4. Therefore, we can represent L with the simple equation

θ = 3π/4

We're told that each radial increment represents π units. The size of the radial increment doesn't have any relevance in this situation. The half of L running in the direction of the angle 7π/4 arises when we allow for all possible negative values of the MPC radius. The answer is B.

5. In Fig. 6-10, line M runs through the origin in the angular direction π/6. Therefore, we can represent M as

θ = π/6

As in the scenario of Question 4, the size of the radial increment doesn't matter here. The half of M running in the direction of the angle 7π/6 arises when we allow for negative radius values. The answer is A.

6. In Fig. 6-10, circle C coincides with the fourth radius circle out from the origin. We're told that each radial increment (distance between any pair of adjacent concentric circles) equals π units. Therefore, circle C has a radius of 4π units. The circle's center lies at the origin, so we know that we can represent C simply with the equation

r = 4π

The angle θ doesn't appear in this equation, because regardless of the value of θ, the value of r equals 4π. The correct choice is A.

7. The airborne object maintains a position at an azimuth bearing of 45, or northeast, while its range (distance from our location at the coordinate origin) equals 28.3 km. We define the direction angle α as the azimuth, or compass, bearing. We call a kilometer a "unit." Therefore, in NPC, our object has the coordinates

(α,r) = (45,28.3)

The answer is B.

8. We can use the NPC-to-Cartesian conversion formulas to solve this problem. We want to convert the NPC ordered pair (α,r) = (45,28.3) to a Cartesian ordered pair (x,y). We calculate

x = r sin α
= 28.3 sin 45
= 28.3 x 0.707
= 20

and

y = r cos α
= 28.3 cos 45
= 28.3 x 0.707
= 20

The Cartesian coordinates are (x,y) = (20,20). The answer is D.

9. We can't "legally" have a negative range coordinate in NPC. The correct choice is D. In each of the other three cases described here, we can "legally" have a negative coordinate value.
10. In the MPC system, we can in fact "legally" have a negative radius value, so we can rule out choice D. Whenever we see a negative radius coordinate in MPC, we can multiply it by -1 and get the equivalent positive radius coordinate going in the opposite direction. In this situation, we have a circle centered at the origin with a radius equivalent to 16 units. The equations r = -16 and r = 16 both represent the same circle in MPC, because the circle goes all the way around, through all possible values of θ (the direction angle). In the Cartesian xy-plane, the equation of a circle centered at the origin with a radius of 16 units is

x2 + y2 = 162

which we normally would write as

x2 + y2 = 256

The correct choice is C.