Trigonometry Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 1
1. The chapter text tells us that two triangles are directly congruent if and only if they're directly similar and the corresponding sides have identical lengths. Direct congruence is a special case of direct similarity. If we find two directly congruent triangles, then we know that they're directly similar. The correct choice is B.
2. A full circle of 360 contains 2π radians. If we consider π = 3.14159, then 2π = 6.28318, so a radian must equal 360 divided by 6.28318. That's approximately 57.3. The correct choice is A.
3. In Fig. 1-15, the uppermost interior angle measures π/2 rad, so it's a right angle. Therefore, we know that the figure is a right triangle. The correct choice is B.
4. In any Euclidean triangle (any triangle on a flat surface) the measures of the interior angles always add up to 180. Therefore, in the situation of Fig. 1-15, we know that x + y = 180 minus the measure of the top angle, which is π/2 rad or 90. By simple arithmetic, x + y = 90, so choice C works. But, because all three angles in any triangle must have measures larger than 0, we can deduce that x < 90 and y < 90, so choices A and B are also correct. The right answer is D, "All of the above."
5. By definition, a triangle is isosceles if and only if two of its sides have the same length. In an equilateral triangle, all three of the sides have the same length, so any equilateral triangle is also isosceles. The correct choice is C.
6. The sine of any angle θ inside a right triangle (other than the right angle) is always positive but less than 1. That is,

0 < sin θ < 1

Choice A represents the correct answer.

7. The cosine of any angle θ inside a right triangle (other than the right angle) is always positive but less than 1. That is,

0 < cos θ < 1

Choice A represents the correct answer.

8. The tangent of any angle θ inside a right triangle (other than the right angle) is always positive. Its value can equal any positive real number; no limit exists as to how large it can get. We write this fact as

tan θ > 0

The correct choice is D.

9. The cosecant of any angle θ inside a right triangle (other than the right angle) equals the reciprocal of the sine of that same angle θ. Because we know that

0 < sin θ < 1

we can conclude that

csc θ > 1

The correct choice is C.

10. A complete circle contains 2π radians. The quantity π/8 equals 1/16 of 2π. Therefore, an angle of π/8 rad represents 1/16 of a full circle. The correct choice is C.