Chapter 13

  1. $ (-)/4 $ 3. $ (-)=-$ 5. a) $ $ b) 1 c) $ (-) $

  2. Hint: $ + + = (+ ) + $ 10. 1 11. $ = -4 ^{3}+ 3 $

  3. $ ^{3}$ 14. $ {4}{2}+1 $

  4. $ (- )^{2} = ^{2}+ ^{2}- 2= 1 - 2= 1 - $ . (The equalities are justified, in order, by algebra, the Pythagorean identity, and the double-angle identity.)

    1. $ ^{2}= (1 - )/2 $ 20. 1/2 ; $ /4 $ 25. d, g, h, i, j, l are true. The rest are false.
    1. 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 $
  1. $ /2 $ n) $ e/$ o) undefined ( $ /e > 1 $ , so it isn’t in arccosine’s domain) p) $ /20 $
  1. b, e, g, h, i are true. The rest are false.

\[ \textcircled{d}\pi/4\quad\textcircled{b})-\pi/4\quad\textcircled{c}0\quad\textcircled{d})-\pi/6\quad\textcircled{e}\pi/3\quad\textcircled{f}\pi/2\quad\textcircled{g})-\pi/3\quad\textcircled{h}0\quad\textcircled{i})-\pi/2\quad\textcircled{j})-\pi/4\quad\textcircled{k})-\pi/9\quad\textcircled{l})-3\pi/14 \]

  1. Odd. 35. Odd. 38. Yes. The first boldface set of solutions includes $ (/3)-18$ (when k=-9).

  2. Both answers are correct: they are different descriptions of the same infinite set of solutions.

  3. For all integers k…

\[ \begin{array}{l l l l l}{a)\left(3\pi/2\right)+2\pi k}&{}&{b)\pm(\pi/4)+2\pi k}&{}&{c)\left(\pi/4\right)+\pi k}&{}&{d)\left(\pi/2\right)+\pi k,\left(\pi/6\right)+2\pi k,\left(5\pi/6\right)+2\pi k}\end{array} \]

    1. $ /3 $ , $ $ , $ 5/3 $ , $ 7/3 $ , $ 3$ , $ 11/3 $ b) $ /4 $ , $ 5/4 $ , $ 9/4 $ , $ 13/4 $
  1. 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