The Angle Mangle

By D. Aviva Rothschild

A sudden stony cold under the left side of my body woke me. "Jeez," I mumbled, rolling onto my back and groping for my blanket--

and I froze in surprise and confusion as my behind pressed into what felt like a concrete floor. Timidly I reached down and confirmed with my fingers that I indeed lay on a concrete floor. Did I fall out of bed? Why don't I hurt? Where's my carpet? Am I dreaming? It sure didn't feel like a dream.

I was afraid to open my eyes, but I forced myself to as I sat up. Instead of the familiar clutter of my room vaguely illuminated by light streaming in through the thin white curtains, I saw a concrete-walled cell lit by a reddish glow, like the red light they use to simulate nocturnal conditions in zoos. Looking wildly around, I didn't see much except a pillow where my head had been lying and a door that looked metallic and heavy, with a huge safe-type lock on it. Something appeared to be moving on the door, but the light was too dim to make it out.

"AH," boomed a bass voice that vibrated through my body, "YOU'RE AWAKE AT LAST."

Now, I'm not one to divide things into good and evil, but that voice personified every nasty thing on Earth. It made me think of televangelists and Hitler and cockroaches. And it scared me down to my toes. "Who are you?" I squeaked. "Where am I?"

"I AM THE PROCTOR. AND YOU ARE IN THE ANGLE TEST. OR, AS IT'S FONDLY KNOWN AROUND HERE, THE ANGLE MANGLE." The Proctor chuckled, a phenomenon I felt more than heard.

I looked around, but unless the Proctor was speaking through the door, I couldn't figure out where he was. The room had no obvious loudspeakers. Finally I asked, "Why am I here?"

The sound of a shrug filled the air. "IT IS YOUR LOT TO BE TESTED."

"Tested?" Some of my fear gave way to annoyance. "On what?"


My fear came surging back. True, I was in the middle of a summer semester class in Trig, but that didn't mean I was ready for a pop quiz.

"CHAPTER 7," the Proctor added.

Oh, God... Frantically I tried to remember everything I'd read. Inverse trig functions and their graphs, trigonometric equations, sum and difference identities, angles of depression and elevation--no, wait, that was chapter 6! Oh God, Oh God, Oh God...

Then sanity (and anger) came flooding in. "Hold it a second," I snapped. "Why am I getting so hyper about this? I'm dreaming, that's all."

"IF I WERE YOU, I WOULDN'T CONTINUE TO BELIEVE THAT," the Proctor said in an offhand manner. I could picture him examining his fingernails.

Uncertainly, I stood up. (Normally I slept naked, but now I was in a nightgown and bathrobe.) I faced the door and said, "Okay. Let's pretend for a moment that this is real. How are you going to test me? What do I have to do? And what happens if I get something wrong?"

"AH." The evil smile in his voice was evident. "YOU MUST ANSWER A QUESTION THAT WILL APPEAR ON THE DOOR. IF YOU ARE CORRECT, THE DOOR WILL OPEN, AND YOU MAY PROCEED DOWN THE HALLWAY TO YOUR NEXT STATION. BUT IF YOU'RE INCORRECT, THE DOOR WILL OPEN, AND SOMETHING WILL HAPPEN." There was enough melodrama and overemphasis in those three words to make William Shatner jealous. "YOU CAN ATTEMPT TO PROCEED TO THE NEXT STATION...." His voice trailed off gleefully.

Oh, God. Test anxiety wasn't usually one of my big problems, but suddenly I dreaded the thought of questions, time limits, failure. It didn't help that I was still fairly sleepy.

"YOU SHOULD GET STARTED," the Proctor said, and I stumbled over to the door. The moving thing I'd seen was a kind of screen saver on a computer screen, because when I got there it flickered into a problem:

g(x) = 7 sec (3/7x+pi/2)-3

Determine domain, range, period, and phase shift. Is this a sinusoid? What are the absolute extrema?

For a horrible few moments I stared with complete incomprehension at the problem. But slowly my brain chugged into gear. This wasn't something completely alien, just a problem like any in my book. Trouble was, had I absorbed enough to remember how to do this?

Do the easy stuff first. "The domain--all real numbers. Except 3/7x +pi/2 can't equal zero. 90, I mean. Pi/2. 3pi/2," I hastily amended. When the Proctor said nothing, I had to conclude that no news was good news, so I plowed ahead: "The range is, uh, -10 to 4. These are also the extrema. The phase shift is, uh, -pi./2 over 3/7, or -7pi/6. The period--" and I stopped, my mind a blank. What was the period of the secant function? Pi? 2pi? The cosine function had a period of 2pi, so it must be 2pi--but what did I do to that number? Put the pi/2 over it? No, then it would lose its pis. "The period is 2pi over 3/7, or 14pi/3. And it is a sinusoid," I concluded nervously.

"MMM," the Proctor grumbled. The door swung open. I cringed, expecting God knew what, but all I saw was a short hallway that opened into another room. I entered the hallway cautiously, but nothing happened, and soon I was in the second room--identical to the first, save that no pillow graced the floor--and, with more confidence, I read the screen in the second door:

Identify these graphs:

The first showed what looked like ivy climbing up the Y-axis. The second was just a wisp of a line that went through the origin. The third was a kind of broad S-curve that also went through the origin.

The first one tipped me off as to the kinds of graphs the Proctor was throwing at me: "That's the inverse sine function." Pointing at the second: "That's the inverse cosine." At the third: "Inverse tangent."

"MMM," the Proctor said, clearly not pleased. The door opened, and I walked confidently in the hall. This wasn't so bad! I remembered more than I'd thought!

On the third door:

Find all solutions to the equation 1 = CSC X- COT X.

This screen came with a light pen that I could draw on the screen with, plus a calculator. "No problem," said I, rubbing my hands together. I wrote down the steps as I converted the equation:

1 = 1/COS X - COS X/SIN X



And suddenly I drew a blank: I couldn't figure out what the best way to proceed was, or even if I'd gone in the right direction in the first place. Long minutes passed while I wrote and rewrote or (more frequently) stared blankly at my scratchings.

Finally the Proctor announced, "TIME. YOU FAIL THIS QUESTION. HA HA HA!"

The door swung silently open.

The hallway looked normal....

But then I smelled it.


I'd read enough mysteries to know--CYANIDE GAS!

I backed up quickly, took a deep breath of the cleanest air possible, and darted forward, hands covering my face. My eyes stung and watered, so that everything blurred away, and the hallway was just long enough to make me desperate to take a breath before I reached the next room, but I managed to hold on until I was well into the next room, where I gasped and wheezed and made desperate fanning motions at the hallway to keep the cyanide out.


Just the thing I needed to hear to restore my calm.

Hoping that the damned Proctor would appear in the flesh at some point so I could strangle him, I checked the fourth door:

Verify the following identity:

Sin2A-Cos2A= 1-2Cos2A

Now this I could do. I wrote:

Sin2A= 1-Cos2A. Thus:

1-Cos2A-Cos2A= 1-2Cos2A

1-2Cos2A= 1-2Cos2A

I stuck my tongue out at the door as it swung open. But my cockiness quickly faded when I gazed upon the fifth door:

Find an exact value for Cos 75

That was all--no graphing utility, no calculator, nothing. When I protested, the Proctor chuckled and said, "YOU MEAN A SMART WOMAN LIKE YOU NEEDS A CALCULATOR FOR SUCH A SIMPLE OPERATION?"

Jerk. Sweating now, I contemplated the test. Obviously, I needed to play with the number, but how? What formula could I use to give me that all-important decimal number?

"DON'T FORGET, I WANT THE EXACT VALUE," the Proctor said as if reading my thoughts.

Well, how much more exact could you get than a calculator-derived decimal? My mind stubbornly refused to become flexible.

Finally I shouted, "All right, so I fail this one, can you at least show me how to do it?"

"OF COURSE," the Proctor replied with the proud magnamanity of a victor to a loser. The words on the door altered:

Cos 75 = Cos (45+30)

= Cos 45*Cos 30-Sin 45*Sin 30

= Ö 2/2*Ö 3/2-Ö 2/2*1/2

= Ö 6/4-Ö 2/4

= (Ö 6-Ö 2)/4

"Oh," I said, feeling like a complete idiot. Of course I'd known how to do that, but panic had completely driven the details away.

Now a new fear loomed as the door swung open. No smell this time, but I could plainly see what the problem was: the hallway was essentially a pit, with just a few inches of "walkway" along the left side. The pit looked bottomless and was certainly nothing I wanted to fall into. So, with infinite care and much sliding of my hands along the wall, I inched along the "walkway," facing the wall rather than the pit.

The floor gave way under my left foot! My hands scrabbled uselessly at the wall--and somehow I managed to hang on, heart beating wildly.

After that I moved even slower, testing each bit of floor as my right foot encountered it. Luckily, the sadistic Proctor had seen fit only to make that one part weak, and I finally made it to the next room.


"Screw you," I panted. My fear was again mutating into anger, which I wasn't sure would make my mind much clearer for the next problem. So this time, before going to the sixth door, I stood in one of the room's corners for a full minute, trying to relax, taking deep breaths, telling myself that I knew what I was doing, that I was learned enough in trig to answer these questions, that the Proctor counted on my fear and confusion to defeat me rather than my ignorance.

That last was a bit of a revelation, and I turned to the sixth door with a surprising amount of calm.

Write the following as an expression involving only Sin X and Cos X:

Sin 2X + Cos 2X

Okay, now what were the double-angle formulas? With my brain functioning normally, I remembered them:

Sin 2X + Cos 2X = 2 Sin X Cos X+Cos²X-Sin²X

"HMMPH," grumbled the Proctor as the door opened.

I didn't say a word or make any gestures as I went to the next room.

Find all solutions in one period of each periodic equation:

4 Sin²X = 3

This one had a calculator and light pen.

Sin²X = 3/4

Sin X = Ö 3/2

Then I merely had to go to the unit circle:

X = pi/3 or X = 2pi/3

"ONE MORE," the Proctor shouted as the door opened. "ONE MORE THAT YOU CAN FAIL."

Well, if I'd rattled him, I must be winning! But I was careful not to get overconfident again, which was a certain path to disaster. Just go and see the final problem and work it and don't think too hard about the future....

Lucky seventh door!

You have a triangle with angle aÐ = 50, side A = 6, and side B = 5. Determine the other angles and side C.

That's funny. I thought the Proctor said this was a test on chapter 7. But this question is from chapter 8. Oh, well, no matter, I had gone well into the laws of sine and cosine. I began to work the problem according to the law of sines....

"WHAT ARE YOU DOING?" shrieked the Proctor in a voice so loud my head nearly fell off.

I winced and said, "Pardon me, but I'm doing a problem here."


"Yes I can. Aren't you up on your chapter 8 methods?"


"Hey," I said huffily, "you never said I had to stick strictly to chapter 7 stuff, just that I had to answer your damn questions."

"NO! NO!" The Proctor's voice got oddly hollow. "YOU CAN'T! YOU CAN'T! You caaaan't...."

The voice faded away. The room spun, although nothing actually moved, so I closed my eyes.

When I opened them again, I was sitting up in bed in my bedroom. It was still night out; my LCD clock said 3:23. My blanket lay on the floor.

Had I been dreaming after all? I was very cold, which I supposed had translated into the feel of cold concrete in my dream, but that was the only clue I had that I might actually have been in the Angle Mangle. Well, that's what I got for studying just before going to bed. "At least it was a good review," I murmured as I scraped up my blanket and settled back to sleep.

Copyright 2001, D. Aviva Rothschild

About the Author

See the "About the Editor" page. This was one of three stories I wrote for a trigonometry class in 1995.

Tell Aviva Rothschild what you thought of her story!

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