|EXPONA SINGS 01|
Wednesday, 25 July 2012
"So," Versine concluded, looking towards the command trig, "Are we done with this enterprise?"
"But we'll be keeping our eyes on you," Cosine added.
"Careful, you'll get a complex," the green-shirted trig shot back, before looking towards Expona. "As for you... pick your battles better. And maybe sing a showtune with ME in it for once!"
Circe blinked and also turned to the redhead. "Expona, you still singing? I thought you were giving it up because it added to the confusion between you and Para."
"Bah! I've decided that the blonde doesn't know musicals," Expona shot back. "So no, I'm staging a return. In fact, someone find me a microphone, I'll demonstrate..."
"You're sure?" Lyn said worriedly.
"Well, rest and food," Nat amended. He glanced at Cotangent. "Have her take two pie and call me in the morning."
Sunday, 22 July 2012
"Well, we can still get Exsecant on a weapons charge," Secant suggested. She pointed to where the function was now being led back to the area, flanked by both Cosecant and Expona.
"On the contrary!" Exsecant cut in. "I was firing my phaser to protect you Trigonometric ratios!"
"How does firing at me protect anything?!" Expona said hotly. "You also shot at Cosecant here too!"
"You looked dangerous!" Exsecant countered. "And with you drawing me away, upon my return I thought Cosecant was a perpetrator fleeing the scene. How was I to know?"
Sine and Cosine exchanged uneasy glances. "Our word against theirs," Cosine yielded. "Just like the last time."
"And the time before that," Sine concurred.
"But this time, you can at least charge Exsecant with impersonation of a security officer," Toc pointed out.
"Aha, yes!" Secant agreed. "Can't explain away that outfit!"
Exsecant glanced down at her yellow-gold top, which would be difficult to pass off as lime green in the current light. "Er, would you believe dry cleaning mixup?" she ventured.
Wednesday, 18 July 2012
As it turned out, when Lyn and the Trigonometric functions emerged from the building, they found that Versine had already been captured. Lyn almost didn't recognize her at first, as the function had removed her earlier accessories and changed into a green top.
"I saw her lurking in the building across the way," Secant remarked, holding Versine by the shoulder. "Oh, and Exsecant took a shot at us, but my sister's got her on the run."
"See how much can get done when I'm in charge?" Circe said smugly. Secant rolled her eyes.
Sine and Cosine exchanged a glance, and each raked their fingers back through their hair before stepping forwards to stare at Versine. "What do you have to say for yourself then?" the twins chorused.
Versine merely smirked. "You have nothing on me. Merely suspicions."
"Objection!" Toc said, lifting an arm high and pointing at the brunette.
Everyone turned to look at him, causing his gaze to turn sheepish as he rubbed the back of his neck. "This would look better if I had a more imposing hairstyle. Or any hairstyle." He cleared his throat. "My point stands though! If you had no hand in this, Versine, why are you here?"
"I have to be somewhere," she shot back. "Despite what the world seems to believe, I still exist!" She put her hands on her hips. "I don't care how much people redefine mathematics, doing silly things like changing the meaning of the word billion... I'm not going away!"
Sine grimaced. "Moreover, she does have a point. We've no proof, and we can't waive her rights."
"Oh, sure, now you stop waving," Circe said, palming her forehead.
Sunday, 15 July 2012
"Ach, s'working!" Cotangent said.
"Wait for it," Cosine cautioned, worriedly watching her two sisters. "Sin(x)/(x) can generate unexpected results..."
Lyn frowned, aware that a part of her also felt connected to what Sine was doing. If something went wrong, might she be harmed too? Shaking her head, she pushed those thoughts aside, focusing on sending positive vibes. Well, positive ones oscillating with negative ones.
Very swiftly, as sinx/x moved further and further along her curve, the explosion got smaller and smaller... until, abruptly, Lyn blinked and it was no longer there. Nor was the TI n-spire, there was merely a charred spot on the table. Sine removed her hand; Tangent didn't immediately disengage the ring though, the strain of her actions still obvious. In fact, it was fortunate that Cotangent was there to catch her, as she collapsed a moment later.
Cosine moved in to support Sine as well, whose features were already shifting back to how she had appeared earlier. "Do they need a medic?" Lyn asked. "Should I run and get Nat?"
"No, we'd better go to him," Cosine asserted. "Worst case, there's a backup device here... best case we can help catch that shifty Archaic reflection of mine."
Sine nodded weakly. "Versine always did have a preference for the merry OLD land of AZ."
Wednesday, 11 July 2012
"Your mind to my mind. Your reciprocal to my amplitude. I am taking your form and combining it with my own..."
With that, it felt like the two fingers that Sine had placed on Lyn's forehead actually slid right inside her head. Lyn gasped despite herself, though Sine's attempt to replicate her didn't really feel painful. In fact, it also allowed Lyn to peer back into Sine's mind.
It was there that Lyn first saw the form of sin(x)/(x). Saw how its maximums and minimums were defined by derivatives. And saw that, while the quotient rule produced the equation (x cos(x) - sin(x))/x^2, upon setting it to zero it would reduce those extremes down to one simple equation. x=tan(x), x not zero. Equating Lyn with her own slope.
Of course, solving that equation would use intersections along y=x and y=tan(x). Where, as each function trended towards infinity, the intersections (and thus max/mins) trended towards Tangent's asymptote values. It was brilliant.
But along with that, Lyn saw more. The blonde realized that the Primary Trig Twins truly did care for Tangent. And that they had not fully realized how hurtful the nickname of Tan-tan, or any of their other teasing, had been in the past. They truly did rely on Tangent for guidance and navigation. And missed her.
Sine's fingers drew back, and when Lyn was able to focus back on the brunette, she saw that the wavy hairdo had been temporarily remolded, and a necklace with Reci's image on it had appeared. "Here we go then," Sine(?) said, exhaling as she turned towards Tangent. "I think either this takes no time at all, or we never quite make it..."
"Zeno would have a field day with us," Cosine remarked.
Sunday, 8 July 2012
Both Tangent and Cotangent reached down to their belts with their free hands. Holding the buckle, they tapped an index finger on the D; it slid off to the side, replaced with an R. The cones from their asymptote rings seemed to shine a bit brighter.
"Go already!" Tangent stated. "Save yourselves, before we run out of energy!"
"Aye! What we're doing, 'tis never been done before!" Cotangent noted.
"Hold on. I have an idea," Sine stated. She tapped at her own belt buckle, shifting it to R, then looked towards Lyn. "Though I'm afraid it involves using you."
Lyn quashed the sudden nervous feelings that Sine's request sparked, as she recalled the conversation at Maud's. "Wh-Whatever I can do to help."
Sine smiled. "My maximums and minimums occur at Tangent's asymptotes. If I use you as a denominator, I'll become a function bounded by your reciprocal... so even though those extremes will offset, they'll continue to diminish in value..."
"Aha! Of course, sin(x)/(x)," Cosine remarked. "Looking like that, Sine may be able to ride the waves off Tangent's asymptote ring, and damp down the explosion using her naturally decreasing amplitude!"
"Ach! What you're saying, 'tis never been done before!" Cotangent protested.
"It's fine, if it saves Tangent along with the rest of you," Lyn cut back in. "Just, what will it involve? You perching on my shoulders or something?"
Sine shook her head. "It would take too long for us to synchronize that way." She took a deep breath. "I'm proposing a mind meld."
There was a pause.
"Well I'm nae repeating myself a third time," Cotangent said in annoyance. "Jest get on wi' it already!"
Wednesday, 4 July 2012
Cosine began to snap her fingers. "There's an idea in that though. Maybe in the future someone can invent a time machine, and come back to save..."
Which was when, at three seconds on the timer, the TI n-Spire suddenly exploded.
Lyn screamed and instinctively shielded her face. So it took a moment or two for her to realize that she hadn't yet been vaporized. "Oh, Tangent," came the twin gasps of both Sine and Cosine.
Lyn quickly turned her attention back to the bomb. Which looked like a small fireball, currently encased in a cone of energy... a cone that was originating in the ring that Tangent wore on her right hand. Which she had pointed at the explosion. Sweat beaded on the brow of the young girl, and she shook, obviously under a great deal of strain.
"Of course! Our asymptote rings!" Cotangent said, looking down at her own hand. "Capable o' causin' something to approach us but nae actually arrive!"
"But eventually the object will be so close that it won't matter!" Sine reminded.
"Oh, rad!" Cosine remarked. There was a pause. "No, seriously, guys, radian mode. The asymptotes could occur more frequently then."
Sunday, 1 July 2012
Happy 145th Birthday, Canada!
It's true, Taylor's Polynomials will be one year old this Tuesday. So we'll be trying out some new things this summer - such as this blog. Don't worry, it will all make sense in context. Once we find out whether the Trigonometry survives...