Trigonometry Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 3
1. Negative angles, by definition, go clockwise. Remember that in radian measure, a full circle equals 2π. If we see an angle of -π/3, we know that its standard equivalent is (2π - π/3) = 5π/3, which represents 5/6 of a complete circle. In degree terms, that's 360 x 5/6 = 300. The correct choice is B.
2. An angle of 855 obviously represents more than one complete circle. In fact, it's two full circles (which would give us 720) plus 135 more. An angle of 135 represents 3/8 of a circle. In radians, that's 2π x 3/8 = 6π/8 = 3π/4. The correct choice is C.
3. By examining the equation, we can tell that the graph has the general contour and shape of a cosine wave. A cosine wave, like a sine wave, is a continuous sinusoidal curve that crosses the independent-variable axis (in this case the x axis) repeatedly. These facts rule out choices B, C, and D (because none of them are false!). However, the statement given in choice A is false. A sinusoid has no singularities. Therefore, the answer is A.
4. According to the unit-circle paradigm in the Cartesian xy-plane, the ordinate of a point on the circle x2 + y2 = 1 represents the sine of the angle counterclockwise from the positive x axis to a ray passing through the circle (assuming that x represents the independent variable and y represents the dependent variable). When we take the reciprocal of the sine, we get the cosecant. The correct choice is D.
5. In the situation described by Question 4, the ratio of the abscissa to the ordinate (or x/y, if we let x represent the independent variable and y represent the dependent variable) equals the cotangent of the angle going counterclockwise from the positive x axis to a ray passing through the unit circle. The correct choice is D.
6. The cosecant function is the reciprocal of the sine function. As we go over the range of angles θ from -6π to 6π, sin θ = 0 whenever θ equals any integer multiple of π. When sin θ = 0, csc θ is undefined; the function "blows up" into a singularity. The correct answer is A. All of the other three choices B, C, and D make false statements, so they don't work here.
7. When we look at a graph of the tangent function y = tan x in the Cartesian xy-plane, we'll see that the output value y can attain any real-number value whatsoever. No matter what value we choose for tan θ, we can always find a point on the graph corresponding to that value. (In fact, we can find infinitely many points on the graph corresponding to any given value of tan θ.) The correct choice is D.
8. When we scrutinize a graph of the sine function y = sin x in the Cartesian xy-plane, we'll see that the output value y can range over the closed interval [-1,1]. The correct choice is A.
9. When we scrutinize a graph of the cosine function y = cos x in the Cartesian xy-plane, we'll see that the output value y can range over the closed interval [-1,1], just as the sine function does (although the horizontal position of the wave differs). The correct choice is A.
10. According to the unit-circle paradigm, we can't define the cosecant function whenever the ordinate (the y value in the xy-plane) equals 0. In those cases, the sine of the angle equals 0, so the cosecant, which is the reciprocal of the sine, "blows up" into a singularity. Of the four choices given in this question, only B represents a situation where y = 0 on the unit circle, so that's the answer.