Math / Helper companion / Trig equations
Uncut · book companion
Kurs 4 trig equations & shifts — our narrative
When two waves meet
An equation f(x) = g(x) is not a magic spell — it asks when two heights agree. Graph both; the crossings are the solutions. Degrees mode on the tool. Later: a wave can sit higher (B) or start early/late (v). Full form A sin k(x+v)+B keeps amplitude, period, bias, and delay in one dial board. (trigonometriska ekvationer · skärningspunkter · förskjutning)
Rebuild “when two heights agree” — then predict the crossings.
Builds on: wave height from the circle (trig functions tool).
Tool gained: crossings of y=f and y=g are solutions of f(x)=g(x).
- Principle. An equation f(x) = g(x) asks when two heights (or two signals) match. On a graph those matches are crossings — each crossing’s x is a solution.
- Model. Plot y = f(x) and y = g(x) (degrees when the chapter uses degrees). Solutions = x-coordinates of the crossings.
- Prediction / Answer. Raise flood line L toward the wave’s peak |A| and meetings slide toward the crests, then vanish above |A|. Read the crossings for a tide or sensor job.
Dial board — stretch, speed, delay, bias.
Builds on: meetings tool above + amplitude/period from trig functions.
Tool gained: dial board y = A sin k(x+v)+B (stretch, speed, delay, bias).
- Principle. The same repeating height can sit higher, start early or late, stretch taller, or wiggle faster — without changing what “a meeting” means.
- Model. y = A sin k(x + v) + B packs four dials: stretch (A), speed (k), delay (v), up/down bias (B). Period still P = 360°/k.
- Prediction / Answer. Slide v and the first crest moves with the clock. Read A, k, and B off a real gauge — same arithmetic, with a why.
1. When two heights agree
A common book move: “draw f and g, click Intersect.” Missing: why a crossing is a solution. Our rewrite: f(x) is one height (or signal); g(x) is another. At an x where the heights match, f(x)=g(x) is true — that x is a solution. The graph is a meeting of two heights, not a button ritual.
Tool/weapon: intersections of graphs solve equations. Plot y=f(x) and y=g(x); each crossing’s x is a solution of f(x)=g(x). Set the calculator (or app) to degrees when the chapter works in degrees.
Worked shape (degrees): solve 2 sin 2x = 1 on 0°…180°. Graph f(x)=2 sin 2x and g(x)=1. Crossings at 15° and 75°. Wider window → more meetings to count.
Example — flood-line crossings
Example — flood-line crossings: graph f(x)=2 sin(2x) and g(x)=1 in degrees. Intersections at 15° and 75° in 0°…180°. Same numbers as the cut “1317”; we gave the tide a job.
Weapon reminder: set digital tools to degrees.
2. Hands-on — watch two traces meet
Drag level L for the flat line g(x)=L against f(x)=2 sin 2x (degrees). Count crossings in 0°…180°. When L=1 you should see the classic pair near 15° and 75°.
L = 1 · crossings in 0°…180° ≈ 2
3. Exercise bank — mark Fake, give Real
Cut-style drills ask you to solve graphically, count solutions, or stare at inequalities with no sensor attached. Below: Fake → Real.
Fake
Fake: “Solve 2 sin 2x = 1.” Naked symbols, no sensor, no stake.
Real — tide vs flood line
Real — tide vs flood line: harbour height ≈ 2 sin(2x) metres (two cycles per half-day map). Flood line sits at 1 m. Crossings at 15° and 75° on that map — two times the water hits the line in the interval. Same arithmetic; now the sensor has a job.
How the fake becomes real
Same equation, same intersections. The real job supplies units (metres) and a reason to care when height equals the flood line.
Fake
Fake: “How many solutions does sin x = cos x have in 0°…360°?” Count without a story.
Real — two sensors agree
Real — two sensors agree: day-length sine and temperature cosine (scaled) read equal. On a full-year / full-circle map they agree twice in 0°…360° (45° and 225°). You care because that’s when the controllers can share one reading.
How the fake becomes real
sin x = cos x ⇔ tan x = 1 (where cos≠0). Two solutions per full turn in the usual open interval — now framed as sensor agreement.
Fake
Fake: “For which positive a does sin x = a have 2 / 1 / 0 solutions in 0°…360°?” — parameter hunt, no gauge.
Real — how often a bob hits a threshold
A balloon height (or scope trace) y=sin x hits threshold a. If |a|<1, twice per loop; if |a|=1, once at the crest/trough; if |a|>1, never — the yank never reaches that line. Same cases as the drill; now a is a tripwire.
How the fake becomes real
Weapon: intersections of y=sin x with y=a. Count depends on whether the line sits inside, on, or outside the amplitude band.
4. Shifted curves — lift and slide
Vertical shift B: y = sin x + B moves the whole wave up (B>0) or down (B<0). Horizontal shift v in the book’s form y = sin(x + v): if v>0 the wave slides left; if v<0 it slides right. (förskjutning i y-led · förskjutning i x-led)
Tool/weapon: P = 360°/k; amplitude |A|; vertical shift B; horizontal shift v (in the book’s form A sin k(x+v)+B, v>0 slides the wave left). Same |A|=(max−min)/2 as before.
Full summary form: y = A sin k(x+v) + B (or cos). |A| = half the peak-to-trough span; P = 360°/k; B is the midline; v is the slide.
Example — read A, k, B from a gauge
Example — read A, k, B: wave peaks at 5, troughs at −1, one loop spans 400°. |A|=(5−(−1))/2=3; B=2; k=360/400=0.9. Form y = 3 sin(0.9x) + 2 (phase aside).
Core: |A|=3, B=2, k=0.9. Same readout as the cut worked example; we named the sensor.
5. Rewrite, max/min, phase stories
To see the slide clearly, rewrite sin(2x + 90°) as sin 2(x + 45°) — factor k out of the argument so the form is k(x+v). Then: sketch the fast wave, slide left by v, then lift by B.
Max / min with bias: for y = A sin(…) + B (or cos), the crest is B+|A| and the trough is B−|A| (when A can be signed, use |A| for the reach).
Fake
Fake: “Read A, k, B from a naked shifted wiggle.” No units, no delay story.
Real — biased, delayed signal
Real — biased, delayed signal: an oscilloscope (or tide gauge) shows a wave from −1 to 5 with one full loop every 400°. Midline B=2, |A|=3, P=400° ⇒ k=360/400=0.9. The bias is a calibration offset; the period sets the filter window.
How the fake becomes real
Same (max−min)/2 and P=360°/k. Balloon / scope hooks welcome — the delay v is “when did the whistle start relative to your clock?”
Fake
Fake: “Victoria claims every sine curve can be written as a cosine. Is she right?” — true/false with no picture of the slide.
Real — same whistle, different clock
Real — same whistle, different clock: cos x = sin(x+90°). Any sine wave is a cosine with a phase shift (and vice versa). On the factory floor that means one sensor curve can be rewritten to match another’s zero-time — same shape, slid along the pipe.
How the fake becomes real
She is right: phase shift renames sin↔cos. The real job is aligning two instruments’ clocks, not winning a true/false point.
Fake
Fake: “The curve y=sin 3x is shifted 36° to the right. Write the new equation.” Symbols only.
Real — delayed radio bob
A radio bob y=sin 3x starts late by 36° of the x-clock (right shift). In the book’s (x+v) form that is v=−36°, so y=sin 3(x − 36°) after factoring carefully — or y=sin(3x − 108°) if you keep the argument packed. Designers need the delay so two antennas’ filters share a phase reference.
How the fake becomes real
Right shift ↔ negative v in their convention. Same algebra; the stake is syncing two whistling feeds.
Gate — four, all required
Hands-on (move L) is 20. Each right gate item is 20. Predict and transfer — do not only recall a button ritual.
1. A crossing of y=f(x) and y=g(x) means…
2. Tide height ≈ 2 sin(2x) m on 0°…180°. Flood line rises from L = 1 to L = 1.8. Predict what happens to the crossings.
3. In the form y = A sin k(x+v) + B, a positive v (book convention) slides the wave…
4. From max=5, min=−1, period=400°, you get…