Chapter 13
$ (-)/4 $ 3. $ (-)=-$ 5. a) $ $ b) 1 c) $ (-) $
Hint: $ + + = (+ ) + $ 10. 1 11. $ = -4 ^{3}+ 3 $
$ ^{3}$ 14. $ {4}{2}+1 $
$ (- )^{2} = ^{2}+ ^{2}- 2= 1 - 2= 1 - $ . (The equalities are justified, in order, by algebra, the Pythagorean identity, and the double-angle identity.)
- $ ^{2}= (1 - )/2 $ 20. 1/2 ; $ /4 $ 25. d, g, h, i, j, l are true. The rest are false.
- 0 b) $ $ c) $ /2 $ d) $ /3 $ e) $ /6 $ f) $ 2/3 $ g) $ 5/6 $ h) $ /3 $ i) $ /7 $ j) $ $ k) $ 8/9 $ l) $ 3/14 $
- $ /2 $ n) $ e/$ o) undefined ( $ /e > 1 $ , so it isn’t in arccosine’s domain) p) $ /20 $
- b, e, g, h, i are true. The rest are false.
\[ \left(\pi/4\quad b\right)-\pi/4\quad c)0\quad d)-\pi/6\quad e)\pi/3\quad f)\pi/2\quad g)-\pi/3\quad h)0\quad i)-\pi/2\quad j)-\pi/4\quad k)-\pi/9\quad l)-3\pi/14 \]
Odd. 35. Odd. 38. Yes. The first boldface set of solutions includes $ (/3)-18$ (when k=-9).
Both answers are correct: they are different descriptions of the same infinite set of solutions.
For all integers k…
\[ \begin{array}{l l l l l}{a)\left(3\pi/2\right)+2\pi k}&{}&{b)\pm(\pi/4)+2\pi k}&{}&{c)\left(\pi/4\right)+\pi k}&{}&{d)\left(\pi/2\right)+\pi k,\left(\pi/6\right)+2\pi k,\left(5\pi/6\right)+2\pi k}\end{array} \]
- $ /3 $ , $ $ , $ 5/3 $ , $ 7/3 $ , $ 3$ , $ 11/3 $ b) $ /4 $ , $ 5/4 $ , $ 9/4 $ , $ 13/4 $
- π/2, π, 3π/2, 2π, 5π/2, 3π, 7π/2, 4π. d) 2π/3, 4π/3, 2π, 8π/3, 10π/3, 4π.
I ndex
Amplitude (of a sinusoidal function), 183
Arccosine, 194
Arclength (of a circular segment), 179
Arcsine, 143, 195
Arctangent, 195
Ass, 94, 133-134
Bartlett, Albert, 115, 118
“Basic Right-Angle Trigonometry”, 149
Beckett, Samuel, 128
Brahmagupta’s Formula, 175
Bretschneider’s Formula, 175
Calvino, Italo, 149, 178
Cancelling (in fractions), 12
Carroll, Lewis, 34
Chiliagon, 132
Circle (equation of), 67
Circumcircle, 73
Cocked Hat, 97
Cofunctions, 148, 161-162
Complementary angles, 131
Completing the square, 13, 50-52, 60, 105-107
Congruence criteria, 136
Congruent triangles, 136
Cooke, Sam, 130
Coordinate axes, 56
Coordinates, 56
Cosecant, 148, 161
Cosine, 145, 147, 156, 163
Cotangent, 148, 161
Concavity, 40
Counting, 3
Cross multiplication, 45
David, 3
Degree (of a polynomial), 36
Delta notation ( $ $ ), 58
Descartes, René, 97, 132, 190
Determining trios (for triangles), 133-134
Devil, 98, 138, 156, 167
Dickinson, Emily, 118, 166
Difference of cubes, 7
Difference of squares, 6
Difference quotient, 82
Discriminant (of a quadratic), 39
Directrix (of a parabola), 108
Distance formula, 66
Distributive property, 3-8, 200
Dividing by zero, 9, 47-48
Domain (of a function), 76
Double-angle identities, 193
Doubling time (of an exponential function), 126
Dummy variable, 79
Einstein, Albert, 10
Equals abuse, 35
Equation (of a graph)
Circle, 67
Defined, 56
Line, 58-62
Parabola, 109
Semicircle, 69
Equations (solving)
Exponential, 122
Linear, 36
Logarithmic, 124
Quadratic, 36-38
Radical, 45-48
Rational, 45
Trigonometric, 196-197
Equivalent equations, 34-35
Eratosthenes, 151
Esau, 5
Even function, 163-164
Exponential function, 114-117
Exponents, 24-27, 31
Factor (of an algebraic expression), 12
Factorial (n!), 14
Factoring (polynomials), 6-7
Feynman, Richard, 77
Fields, W.C., 92
Fitzgerald, Ella, 152
Fish curve, 98
Flag angles, 131
Focus (of a parabola), 108
FOIL, 5
Folium of Descarte, 97
Fractions, 11-13, 16-17
Function (definition and notation), 76-77
Function of two variables, 99
Parentheses, 15, 79
Fundamental Principle of Coordinate Geom., 56
Parker, Dorothy, vi
Galilei, Galileo, 108
Pentagram, 138
Gelfand, I.M., vi, vii
Period (of a trigonometric function), 180
Gilbert, W.S. 105
Perpendicular bisector, 73
Golden ratio, 138
Perpendicular lines (slopes), 62
Gradian (angle measure), 179
Plato, 3
Grasshoppers, 67
Poincaré, Henri, 161
Hendrix, James Marshall, 49
Point-slope formula, 59
Heron’s formula, 172-174
Polynomial, 36
Hertz, Heinrich, 26
Pooka, 212
Hollander, John, 92
“Pre-negative” quadratic formula, 54
Holmes, Sherlock, 144, 147
Pythagorean Identity, 163
Intersections, 71, 74
Pythagorean Theorem, 64, 65, 139
Inverse functions (definition), 120
Quadratic Formula, 37, 51
Inverse trigonometric functions, 143, 194–195
Quadratic polynomials (graphs), 105-106
Jacob, 5
Radians, 178
Joyce, James, 126
Radical expression, 45
Kafka, Franz, 92
Radicals, 29
Kepler, Johannes, 64
Range (of a function), 76
Kronecker, Leopold, 3
Rational expression, 45
Lang, Serge, vii
Rationalizing the denominator, 32
Law of Cosines, 167
Reciprocal, 14
Law of Sines, 166
Reciprocal trigonometric functions, 148
Libby, Willard, 128
Reflection. 92
Lines (in coordinate geometry), 58-62
Rhombus, 136
Logarithms, 118-125
Rice, 114
Long Shu, 114
Roots, 29
Macbeth, 93
“Rule of 70”, 127
Marx Brothers, 53
SAS Area formula, 171
Melville, Hermann, 133, 153
Secant, 148, 161
Minus times minus is plus, 8
Semicircles, 69
Monomial, 36
Shakespeare, William, 93
Montesquieu, 171
Sheba, Queen of, 10
Moses, 95, 144
Shifts. 94
Multiply-by-1 Trick, 13, 16
Similarity criterion (AA), 137
Nabokov, Vladimir, 122, 160
Similar triangles, 137
Nancy, 170
Sine, 142, 145, 156, 163
Natural logarithm, 118-124
Sine wave, 158
Naughty, naughty, 44 (#24)
Slope, 58, 62
Negative numbers, 53
Socrates, 3
Odd function. 163-164
SOH CAH TOA, 146, 156, 160, 183
Origin, 56
Solomon, 10, 154
Parabolas, 108-112
Spengler, Oswald, 76
Parallel lines (slopes), 63
Sterne, Laurence, 61
Parallelogram (area), 64
Stretches, 93
Sum identities (for sine and cosine), 190-191
Supplementary angles, 131
System of equations, 71-74
Tangent (trig function), 145, 147, 160, 163
Taylor polynomial, 188
Tennyson, Alfred Lord, 12
Term (of an algebraic expression), 12
Track covering, 35
Transformations, 92
Transformation Table, 95, 98, 101
Unit circle. 67. 156. 160
“Unnatural logarithms”, 125
Vertical angles, 131
Washington, George, 94
Whitehead, Alfred North, 24
Whores, vi, 118
Wicked Bible, 95
Xu Fu, 181
YHWH, 3
Zero equals one, 48
Zero Product Theorem, 37, 47
Zigzag angles, 131