| Introduction |
|
1 | (4) |
|
|
|
1 | (1) |
|
|
|
2 | (1) |
|
|
|
3 | (1) |
|
|
|
3 | (1) |
|
|
|
4 | (1) |
|
PART 1 GETTING STARTED WITH TRIGONOMETRY |
|
|
5 | (84) |
|
Chapter 1 Taking On Trig Technicalities |
|
|
7 | (24) |
|
Taking Trig for a Ride: What Trig Is |
|
|
7 | (1) |
|
Sizing up the basic figures |
|
|
8 | (2) |
|
Identifying angles and their names |
|
|
10 | (2) |
|
Taking on triangles and their angles |
|
|
12 | (1) |
|
Going outside the triangle |
|
|
13 | (1) |
|
Making a circle work from every angle |
|
|
13 | (3) |
|
Looking at angles in a circle |
|
|
16 | (3) |
|
|
|
19 | (1) |
|
Making the words fit the triangle |
|
|
19 | (3) |
|
Making triangles less radical |
|
|
22 | (2) |
|
|
|
24 | (2) |
|
Finding Trig Applications in the Basics |
|
|
26 | (1) |
|
|
|
26 | (1) |
|
|
|
27 | (1) |
|
|
|
28 | (1) |
|
|
|
29 | (2) |
|
Chapter 2 Cooperating with Cartesian Coordinates |
|
|
31 | (20) |
|
Starting Out Simple: Plotting Points |
|
|
31 | (1) |
|
Axes, axes, we all fall down |
|
|
32 | (1) |
|
Determining the origin of it all |
|
|
32 | (1) |
|
|
|
32 | (1) |
|
Cutting the graph into four parts |
|
|
33 | (1) |
|
From Here to There: Calculating Distances |
|
|
34 | (1) |
|
Counting on vertical and horizontal distances |
|
|
34 | (1) |
|
Another slant: Diagonal distances |
|
|
35 | (2) |
|
Using exact values or estimating distances |
|
|
37 | (1) |
|
Getting to the Center of It All |
|
|
37 | (1) |
|
Finding the midpoint of a line segment |
|
|
38 | (1) |
|
Locating the center of a circle |
|
|
38 | (2) |
|
Partitioning line segments further |
|
|
40 | (2) |
|
Pinpointing the center of a triangle |
|
|
42 | (2) |
|
|
|
44 | (1) |
|
|
|
44 | (1) |
|
Recognizing parallel and perpendicular lines |
|
|
45 | (1) |
|
Defining Circles with Numbers |
|
|
46 | (1) |
|
Centering circles at the origin |
|
|
46 | (1) |
|
|
|
47 | (1) |
|
Circling Around with Applications |
|
|
47 | (1) |
|
|
|
47 | (2) |
|
|
|
49 | (2) |
|
Chapter 3 Finding Degrees in Triangles and Planes |
|
|
51 | (12) |
|
Angles, Angles Everywhere: Measuring in Degrees |
|
|
51 | (1) |
|
Slicing a coordinate plane |
|
|
52 | (1) |
|
Looking elsewhere for degree measures |
|
|
52 | (4) |
|
Graphing Angles in Standard Position |
|
|
56 | (1) |
|
Positioning initial and terminal sides |
|
|
56 | (1) |
|
|
|
56 | (1) |
|
What's Your Angle? Labeling in Various Ways |
|
|
57 | (1) |
|
Using negative angle measures |
|
|
57 | (1) |
|
Comingling with coterminal angles |
|
|
57 | (2) |
|
Renaming angles: So many aliases |
|
|
59 | (1) |
|
Making Degrees Work for You |
|
|
60 | (1) |
|
|
|
60 | (1) |
|
|
|
61 | (2) |
|
Chapter 4 Dishing Out the Pi: Radians |
|
|
63 | (14) |
|
|
|
63 | (1) |
|
|
|
64 | (1) |
|
Converting degrees and radians |
|
|
65 | (3) |
|
|
|
68 | (1) |
|
|
|
68 | (1) |
|
Taking chunks out of circles |
|
|
69 | (3) |
|
|
|
72 | (2) |
|
|
|
74 | (3) |
|
Chapter 5 Tackling Right Triangles |
|
|
77 | (12) |
|
Sizing Up Right Triangles |
|
|
77 | (1) |
|
What's so right about them? |
|
|
78 | (1) |
|
The anatomy of a right triangle |
|
|
78 | (2) |
|
Demystifying the Pythagorean Theorem |
|
|
80 | (1) |
|
Hitting a Pythagorean triple |
|
|
80 | (1) |
|
Solving for a missing length |
|
|
81 | (3) |
|
In a League of Their Own: Special Right Triangles |
|
|
84 | (1) |
|
|
|
84 | (1) |
|
Isosceles right triangles |
|
|
85 | (1) |
|
Getting the Applications Right |
|
|
86 | (1) |
|
|
|
86 | (1) |
|
|
|
87 | (2) |
|
PART 2 TRIGONOMETRIC FUNCTIONS |
|
|
89 | (70) |
|
Chapter 6 Describing Trig Functions |
|
|
91 | (16) |
|
Discovering How Trig Functions Work |
|
|
92 | (1) |
|
The name game: A right triangle's three sides |
|
|
92 | (1) |
|
The six ratios: Relating the three sides |
|
|
92 | (1) |
|
The sine function: Opposite over hypotenuse |
|
|
93 | (1) |
|
The cosine function: Adjacent over hypotenuse |
|
|
94 | (1) |
|
The tangent function: Opposite over adjacent |
|
|
95 | (2) |
|
All together, now: Using one function to solve for another |
|
|
97 | (1) |
|
Similar right triangles within a right triangle |
|
|
97 | (1) |
|
Taking It a Step Further: Reciprocal Functions |
|
|
98 | (1) |
|
The cosecant function: Sine flipped upside down |
|
|
99 | (1) |
|
The secant function: Cosine on its head |
|
|
100 | (1) |
|
The cotangent function: Tangent, tails side up |
|
|
100 | (1) |
|
Angling In on Your Favorites |
|
|
101 | (1) |
|
Identifying the most popular angles |
|
|
101 | (1) |
|
Determining the exact values of functions |
|
|
102 | (3) |
|
|
|
105 | (2) |
|
Chapter 7 Relating Triangles to Circular Functions |
|
|
107 | (16) |
|
Getting Acquainted with the Unit Circle |
|
|
108 | (1) |
|
Placing points on the unit circle |
|
|
108 | (3) |
|
Finding a missing coordinate |
|
|
111 | (1) |
|
Sticking to rational coordinates |
|
|
112 | (2) |
|
Going Full Circle with the Angles |
|
|
114 | (1) |
|
|
|
114 | (1) |
|
Being negative or multiplying your angles |
|
|
115 | (1) |
|
Locating and computing reference angles |
|
|
116 | (3) |
|
Navigating with Circular Measures |
|
|
119 | (1) |
|
|
|
119 | (1) |
|
Cycling with a cyclic quadrilateral |
|
|
120 | (3) |
|
Chapter 8 Taking Trig Functions Global |
|
|
123 | (16) |
|
Defining Trig Functions for All Angles |
|
|
123 | (1) |
|
Putting reference angles to use |
|
|
124 | (1) |
|
Labeling the optimists and pessimists |
|
|
124 | (1) |
|
|
|
125 | (2) |
|
Using Coordinates of Circles to Solve for Trig Functions |
|
|
127 | (1) |
|
Calculating with coordinates on the unit circle |
|
|
128 | (1) |
|
Calculating with coordinates on any circle at the origin |
|
|
129 | (2) |
|
Defining Domains and Ranges of Trig Functions |
|
|
131 | (2) |
|
Friendly functions: Sine and cosine |
|
|
133 | (1) |
|
Close cousins of their reciprocals: Cosecant and secant |
|
|
133 | (1) |
|
Brothers out on their own: Tangent and cotangent |
|
|
134 | (1) |
|
Applying the Trig Functions |
|
|
135 | (1) |
|
Flying around on a Ferris wheel |
|
|
135 | (1) |
|
Trying out some new trig functions |
|
|
136 | (3) |
|
Chapter 9 Applying Yourself to Trig Functions |
|
|
139 | (20) |
|
First Things First: Elevating and Depressing |
|
|
139 | (2) |
|
Measuring Tall Buildings with a Single Bound |
|
|
141 | (1) |
|
Rescuing a child from a burning building |
|
|
141 | (2) |
|
Determining the height of a tree |
|
|
143 | (1) |
|
Measuring the distance between buildings |
|
|
144 | (1) |
|
|
|
145 | (2) |
|
The Sky's (Not) the Limit |
|
|
147 | (1) |
|
|
|
148 | (2) |
|
|
|
150 | (1) |
|
Measuring the view of satellite cameras |
|
|
151 | (2) |
|
Calculating Odd Shapes and Maneuvering Corners |
|
|
153 | (1) |
|
Finding the area of a triangular piece of land |
|
|
153 | (2) |
|
|
|
155 | (1) |
|
Moving an object around a corner |
|
|
155 | (4) |
|
|
|
159 | (72) |
|
Chapter 10 Introducing Basic Identities |
|
|
161 | (16) |
|
Flipping Functions on Their Backs: Reciprocal Identities |
|
|
162 | (1) |
|
Function to Function: Ratio Identities |
|
|
163 | (1) |
|
Opposites Attract: Opposite-Angle Identities |
|
|
164 | (3) |
|
Revisiting the Classic Theorem: Pythagorean Identities |
|
|
167 | (1) |
|
The mother of all Pythagorean identities |
|
|
168 | (1) |
|
Extending to tangent and secant |
|
|
169 | (1) |
|
Finishing up with cotangent and cosecant |
|
|
170 | (1) |
|
Rearranging the Pythagorean identities |
|
|
171 | (2) |
|
|
|
173 | (1) |
|
|
|
173 | (1) |
|
|
|
174 | (3) |
|
Chapter 11 Operating on Identities |
|
|
177 | (20) |
|
|
|
177 | (5) |
|
Overcoming the Differences |
|
|
182 | (3) |
|
|
|
185 | (1) |
|
One plus one equals two sines |
|
|
186 | (2) |
|
|
|
188 | (2) |
|
|
|
190 | (1) |
|
|
|
191 | (1) |
|
Half a tangent is double the fun |
|
|
191 | (1) |
|
Using half-angle identities |
|
|
192 | (2) |
|
Comparing Exact Values and Estimations |
|
|
194 | (3) |
|
Chapter 12 Proving Identities: The Basics |
|
|
197 | (16) |
|
|
|
198 | (1) |
|
|
|
199 | (4) |
|
|
|
203 | (2) |
|
|
|
205 | (1) |
|
Changing to sines and cosines |
|
|
206 | (3) |
|
|
|
209 | (1) |
|
Using a little bit of both |
|
|
210 | (3) |
|
Chapter 13 Sleuthing Out Identity Solutions |
|
|
213 | (18) |
|
|
|
213 | (1) |
|
Breaking up is hard to do |
|
|
214 | (2) |
|
Finding a common denominator |
|
|
216 | (3) |
|
Using Tricks of the Trig Trade |
|
|
219 | (1) |
|
Multiplying by a conjugate |
|
|
219 | (2) |
|
|
|
221 | (1) |
|
Identifying with the Operations |
|
|
222 | (1) |
|
|
|
223 | (1) |
|
What difference does it make? |
|
|
224 | (2) |
|
|
|
226 | (1) |
|
Halving fun, wish you were here |
|
|
227 | (2) |
|
Applying the Magic of Trigonometry |
|
|
229 | (1) |
|
Making some given information work |
|
|
229 | (1) |
|
|
|
229 | (2) |
|
PART 4 EQUATIONS AND APPLICATIONS |
|
|
231 | (70) |
|
Chapter 14 Investigating Inverse Trig Functions |
|
|
233 | (8) |
|
|
|
233 | (1) |
|
|
|
234 | (1) |
|
Distinguishing between the few and the many |
|
|
235 | (2) |
|
Determining Domain and Range of Inverse Trig Functions |
|
|
237 | (1) |
|
|
|
238 | (1) |
|
|
|
238 | (1) |
|
|
|
238 | (1) |
|
Inverse cotangent function |
|
|
238 | (1) |
|
|
|
239 | (1) |
|
Inverse cosecant function |
|
|
239 | (1) |
|
Summarizing domain and range |
|
|
239 | (2) |
|
Chapter 15 Making Inverse Trig Work for You |
|
|
241 | (12) |
|
|
|
241 | (2) |
|
Getting Friendly with Your Calculator |
|
|
243 | (1) |
|
|
|
244 | (1) |
|
Interpreting notation on the calculator screen |
|
|
244 | (3) |
|
|
|
247 | (1) |
|
Solving Some Mixed Problems |
|
|
248 | (2) |
|
|
|
250 | (3) |
|
Chapter 16 Solving Trig Equations |
|
|
253 | (24) |
|
Generating Simple Solutions |
|
|
254 | (1) |
|
Factoring In the Solutions |
|
|
255 | (1) |
|
Finding a greatest common factor |
|
|
256 | (1) |
|
|
|
257 | (2) |
|
Increasing the degrees in factoring |
|
|
259 | (3) |
|
|
|
262 | (1) |
|
Using the Quadratic Formula |
|
|
263 | (1) |
|
|
|
264 | (4) |
|
Finding Multiple-Angle Solutions |
|
|
268 | (2) |
|
|
|
270 | (2) |
|
|
|
272 | (1) |
|
Solving with a Graphing Calculator |
|
|
273 | (4) |
|
Chapter 17 Obeying the Laws and Applying Them |
|
|
277 | (24) |
|
Describing the Parts of Triangles |
|
|
278 | (1) |
|
|
|
278 | (1) |
|
|
|
278 | (2) |
|
Following the Law of Sines |
|
|
280 | (4) |
|
Continuing with the Law of Cosines |
|
|
284 | (1) |
|
Defining the law of cosines |
|
|
284 | (1) |
|
|
|
285 | (2) |
|
|
|
287 | (2) |
|
|
|
289 | (4) |
|
Finding the Areas of Triangles |
|
|
293 | (1) |
|
Finding area with base and height |
|
|
294 | (1) |
|
Finding area with three sides |
|
|
295 | (2) |
|
|
|
297 | (1) |
|
|
|
298 | (3) |
|
PART 5 THE GRAPHS OF TRIG FUNCTIONS |
|
|
301 | (52) |
|
Chapter 18 Graphing Sine and Cosine |
|
|
303 | (16) |
|
|
|
303 | (1) |
|
|
|
304 | (1) |
|
Describing amplitude and period |
|
|
305 | (2) |
|
Formalizing the sine equation |
|
|
307 | (1) |
|
|
|
307 | (3) |
|
|
|
310 | (1) |
|
|
|
310 | (1) |
|
Using properties to graph cosine |
|
|
311 | (1) |
|
Applying the Sines of the Times |
|
|
311 | (1) |
|
|
|
312 | (1) |
|
|
|
313 | (1) |
|
|
|
314 | (2) |
|
|
|
316 | (1) |
|
Theorizing with biorhythms |
|
|
316 | (3) |
|
Chapter 19 Graphing Tangent and Cotangent |
|
|
319 | (8) |
|
|
|
319 | (1) |
|
|
|
320 | (1) |
|
|
|
320 | (1) |
|
Fiddling with the tangent |
|
|
321 | (3) |
|
Confronting the Cotangent |
|
|
324 | (3) |
|
Chapter 20 Graphing Two More Trig Functions |
|
|
327 | (12) |
|
Seeing the Cosecant for What It Is |
|
|
327 | (1) |
|
Identifying the asymptotes |
|
|
328 | (1) |
|
|
|
328 | (1) |
|
|
|
329 | (2) |
|
|
|
331 | (1) |
|
Determining the asymptotes |
|
|
331 | (1) |
|
Sketching the graph of secant |
|
|
332 | (1) |
|
Fooling around with secant |
|
|
333 | (2) |
|
Laying Out the Inverse Functions |
|
|
335 | (1) |
|
Graphing inverse sine and cosine |
|
|
335 | (1) |
|
Taking on inverse tangent and cotangent |
|
|
336 | (1) |
|
Crafting inverse secant and cosecant |
|
|
337 | (2) |
|
Chapter 21 Topping Off Trig Graphs |
|
|
339 | (14) |
|
The Basics of Trig Equations |
|
|
339 | (2) |
|
Flipping over a horizontal line |
|
|
341 | (1) |
|
Interpreting the equation |
|
|
341 | (1) |
|
Graphing with the General Form |
|
|
342 | (4) |
|
Adding and Subtracting Functions |
|
|
346 | (2) |
|
Applying Yourself to the Task |
|
|
348 | (1) |
|
|
|
348 | (1) |
|
Tracking the deer population |
|
|
349 | (2) |
|
Measuring the movement of an object on a spring |
|
|
351 | (2) |
|
|
|
353 | (14) |
|
Chapter 22 Ten Basic Identities ... Plus Some Bonuses |
|
|
355 | (6) |
|
|
|
355 | (1) |
|
|
|
356 | (1) |
|
Checking in with the cosine |
|
|
356 | (1) |
|
Off on a tangent with its reciprocal |
|
|
356 | (1) |
|
|
|
357 | (1) |
|
Creating the ratio identity for tangent |
|
|
357 | (1) |
|
Making the cotangent a ratio identity |
|
|
357 | (1) |
|
Pythagorean Identity Plus |
|
|
358 | (1) |
|
Opposite-Angle Identities |
|
|
358 | (1) |
|
Multiple-Angle Identities |
|
|
359 | (1) |
|
|
|
359 | (1) |
|
|
|
359 | (1) |
|
Tangent keeps its fractional origin |
|
|
359 | (2) |
|
Chapter 23 Ten Not-So-Basic Identities |
|
|
361 | (6) |
|
Product-to-Sum Identities |
|
|
361 | (2) |
|
Sum-to-Product Identities |
|
|
363 | (1) |
|
|
|
364 | (1) |
|
|
|
364 | (3) |
| Appendix: Graphs And Function Values |
|
367 | (6) |
| Index |
|
373 | |
