Pure Mathematics 1

Deep Dive: Trig Graphs Period and Domain

Provenance: written 2026-09-29. Sources actually read for this document: research/veridian/WMA11-verified-facts.md (§1 spec table of contents; §4.1 "Graph transformations"; §4.3 "Trigonometry"; §6; §7; the trig-graphs-period-and-domain.ts lesson-audit-log entry, lines ~1082–1189) — grep'd and read in full for every "trig", "period", "tan", "domain" and "reflect" hit; the live lesson file content/veridian/curriculum/wma11/trig-graphs-period-and-domain.ts (all 908 lines, read in full, to avoid duplicating its worked chains, diagrams and MCQs); content/veridian/technique-hub/wma11/codex-formula-and-method-bank.md §1.3 and §2.3; content/veridian/technique-hub/wma11/forge-exam-technique-guide.md §4.3; content/veridian/technique-hub/wma11/signal-frequency-and-boundaries.md in full; and content/veridian/question-bank/wma11.ts (confirmed empty of any trig content — its two entries are both spec 1.3/1.6). No new primary-source PDF was fetched for this pass; every citation below is inherited from the facts bank's own already-verified extractions, re-checked here only for correct attribution and scope. All period/domain/symmetry identities stated below as general theory were independently checked numerically (Python, exact and floating-point comparison against math.sin/cos/tan) before being written in, not asserted from memory. Correction pass, same date: §6 item 3 originally followed the facts bank's own §4.1 framing of Oct 2021 Q4(a) as a "describe the transformation" question; cross-checking against this same folder's graph-transformations-in-words.ts codex deep-dive and SIGNAL brief and this topic's own SIGNAL brief — all three independently re-verified against the primary paper — showed the real question instead asks for a minimum point's coordinates and a period-derived value of k, so §6 item 3 has been rewritten to match and to flag the correction explicitly.

#CODEX Deep-Dive — Trig Graphs: Period and Domain (WMA11, spec 3.3)

#0. Spec anchor, verbatim, and where this document's boundary sits

Spec 3.3 (Issue 3, April 2019, p.14) covers sine, cosine and tangent functions — their graphs, "symmetries and periodicity," per the spec's own heading — and names three functions as expected knowledge: $y=3\sin x$ , $y=\sin(x+\pi/6)$ , $y=\sin 2x$ (paraphrased from the facts bank's spec extraction, §1). Three words carry the whole content list — graphs, symmetries, periodicity — and the three named examples are Pearson's own, not invented for teaching purposes.

Spec 1.12 is the general transformation rule this topic borrows its mechanism from: four named forms — $y=af(x)$ , $y=f(x)+a$ , $y=f(x+a)$ , $y=f(ax)$ — which the spec states are, verbatim, "applicable to... sine, cosine and tangent" among other base functions (paraphrased/short-quoted from the facts bank's spec extraction, §1). Four transformation types, stated once and applied across every base function the spec lists — sine, cosine and tangent included by name.

This document does not re-derive what the live lesson (trig-graphs-period-and-domain.ts) already builds in full: the tan x period worked-chain (Jan 2023 Q9(a)), the sin 2x chain-drill, the four diagrams, or the six MCQs. It exists to go one level further than any single worked example can: the general form of each rule the lesson only shows for one specific coefficient, the one spec-1.12 transformation type (vertical translation) that is absent from every existing WMA11 document on this topic, and an honest accounting of exactly which real exam angles on this exact spec point exist in the verified-facts corpus versus which are spec-required theory with no confirmed citation.

#1. Definitions, stated precisely

Period. A function $f$ is periodic if some positive $T$ satisfies $f(x+T) = f(x)$ for every $x$ in its domain. The period is the smallest such $T$ . This matters beyond vocabulary: if $T$ works, so does every positive integer multiple $nT$ ( $f(x+nT) = f(x)$ follows by applying $f(x+T)=f(x)$ repeatedly) — so a correct-but-not-minimal repeat-length is a real number that genuinely returns the graph to itself, and is still a wrong answer to "state the period." $4\pi$ for $\sin x$ is the standard example the live lesson already uses; the same logic applies to $tan x$ and $n\pi$ for any integer $n \geq 1$ .
Domain. The set of $x$ -values at which $f$ is defined — for $\sin x$ and $\cos x$ , all reals; for $\tan x$ , all reals except $x = \tfrac{\pi}{2} + n\pi$ . A set, not a length; comparing a domain to a period is a category error even when, as with $\tan x$ 's single-branch domain width, the two happen to share the same numeral.
Range. The set of $y$ -values actually output — $[-1,1]$ for $\sin x$ and $\cos x$ ; all reals for $\tan x$ (unbounded, hence no amplitude).
Amplitude. Half the distance between a bounded periodic function's maximum and minimum output — $1$ for $\sin x$ and $\cos x$ ; undefined for $\tan x$ .
Symmetry, spec 3.3's own third word, has (at minimum) two independent senses relevant here, and only the first is covered by the live lesson:
- Global symmetry of the whole graph: $\sin x$ and $\tan x$ are odd ( $f(-x)=-f(x)$ , point symmetry about the origin); $\cos x$ is even ( $f(-x)=f(x)$ , line symmetry about the $y$ -axis).
- Local (within-period) symmetry: every sine/cosine graph also has a vertical mirror line through each of its peaks and troughs, and a point of rotational symmetry at each zero — a genuinely separate fact from the origin/y-axis symmetry above, and one no existing WMA11 document on this topic states. §3.5 below gives it precisely.

#2. The mechanism, generalised beyond a single coefficient

The live lesson derives, for the one case $k=2$ , why $\tan x$ 's period is $\pi$ : $\sin(x+\pi)=-\sin x$ and $\cos(x+\pi)=-\cos x$ (both flip sign under a half-turn), so the two minus signs cancel in the ratio $\tan(x+\pi) = \sin(x+\pi)/\cos(x+\pi) = \tan x$ , and no smaller shift produces an equivalent cancellation. That argument does not depend on which specific multiple of $x$ is inside the function — it generalises cleanly, and the general form is the actual spec-1.12 content (" $y=f(ax)$ ... applicable to... tangent") the single worked case is one instance of.

General period rule, $k \neq 0$ : if $f$ has minimal period $P$ , then $g(x) = f(kx)$ has minimal period $P/|k|$ . Why the minimality survives, not just the repeat itself: $g(x + P/|k|) = f(kx \pm P) = f(kx) = g(x)$ , so $P/|k|$ works; if some smaller $T' < P/|k|$ also worked for $g$ , then $f$ would repeat after the shift $|k|T' < P$ , contradicting $P$ 's own minimality for $f$ . So the scaling carries the smallest label across, not merely a working shift.

$\sin(kx)$ , $\cos(kx)$ : period $\dfrac{2\pi}{|k|}$ .
$\tan(kx)$ : period $\dfrac{\pi}{|k|}$ , domain $x \neq \dfrac{\pi}{2k} + \dfrac{n\pi}{k}$ for integer $n$ (from $\cos(kx)=0 \iff kx = \tfrac{\pi}{2}+n\pi$ ).

Micro-example A (stretch smaller than 1, the direction the live lesson never shows — it only covers $k=2$ ): $y = \cos\!\left(\tfrac{x}{2}\right)$ has $k = \tfrac{1}{2}$ , so period $= 2\pi \div \tfrac{1}{2} = 4\pi$ — a horizontal stretch, scale factor $2$ , not a compression, doubling the period rather than halving it. The general rule (scale factor $1/k$ ) runs in both directions; a lesson built around only $k=2$ leaves the $k<1$ direction untested.

Micro-example B (tangent, a coefficient the live lesson never applies to tan x at all): $y = \tan(3x)$ has period $\pi/3$ and its first positive asymptote at $x = \pi/6$ (from $3x = \pi/2$ ), with the full domain restriction $x \neq \tfrac{\pi}{6} + \tfrac{n\pi}{3}$ . Checked directly: $\cos(3 \cdot \pi/6) = \cos(\pi/2) = 0$ . ✓.

#3. The four spec-1.12 transformations, systematically, on sin/cos/tan

Form	Acts on	Effect on a trig graph	Covered by the live lesson?
$y = af(x)$	output	Vertical stretch × $a$ : amplitude → $\lvert a\rvert$ , range → $[-\lvert a\rvert, \lvert a\rvert]$ for sin/cos; period unchanged. If $a<0$ , also a reflection in the $x$ -axis.	Yes ( $3\sin x$ ), but only $a>0$ — no reflection case.
$y = f(x) + a$	output	Vertical translation by $a$ : range shifts by $a$ (e.g. $[-1,1] \to [-1+a, 1+a]$ for sin/cos); amplitude and period both unchanged.	No — absent from the lesson, the formula bank and the forge guide alike.
$y = f(x+a)$	input	Horizontal translation by $-a$ (i.e. left by $a$ when $a>0$ ); amplitude, period, range all unchanged.	Yes ( $\sin(x+\pi/6)$ ), but only the leftward ( $a>0$ ) case.
$y = f(ax)$	input	Horizontal stretch × $1/a$ : period → (original period) $/\lvert a\rvert$ ; amplitude/range unchanged for sin/cos. If $a<0$ , also a reflection in the $y$ -axis.	Yes ( $\sin 2x$ ), but only $a=2$ , and no reflection case.

Micro-example C — the missing transformation, $f(x)+a$ : $y = \cos x + 1$ . Amplitude stays $1$ , period stays $2\pi$ ; the range shifts from $[-1,1]$ to $[0,2]$ . Concretely: the curve's minimum, previously $-1$ at $x=\pi$ , is now $\cos(\pi)+1 = 0$ — the graph touches but does not cross the $x$ -axis at its lowest point, rather than dipping below it. This is spec-1.12 content applied to a trig function exactly as the spec itself permits (" $y=f(x)+a$ ... applicable to... cosine"), and it is genuinely untaught anywhere else in this course's WMA11 material on this topic.

Micro-example D — reflection as a sub-case of $af(x)$ with $a<0$ : $y = -\sin x$ . Not a new transformation rule — it is $af(x)$ with $a=-1$ — but the shape-change is easy to misdescribe as something else: every peak becomes a trough and vice versa (reflection in the $x$ -axis), while period ( $2\pi$ ) and amplitude ( $1$ ) are both untouched. Because $\sin x$ is odd, $-\sin x = \sin(-x)$ too — for this one function, a reflection in the $x$ -axis and a reflection in the $y$ -axis produce an identical graph, which is itself a direct consequence of the odd-symmetry fact in §1, not a coincidence. $\cos x$ , being even, does not share this shortcut: $-\cos x \neq \cos(-x) \, (= \cos x)$ — its $x$ -axis and $y$ -axis reflections are genuinely different graphs.

Micro-example E — a combined transformation using two of the spec's own three named examples' mechanisms at once: $y = 3\sin(2x)$ combines $af(x)$ ( $a=3$ , amplitude → 3) with $f(ax)$ ( $a=2$ , period → $\pi$ ). Because one acts purely on the output and the other purely on the input, they don't interact or require a specific order: amplitude 3, period $\pi$ , range $[-3,3]$ , unchanged from applying either transformation to the other's result.

#4. A harder combined sub-case: two input-side transformations, where order genuinely matters

Spec 1.12 lists $f(x+a)$ and $f(ax)$ as separate transformations, but both act on the input — and unlike the amplitude/period pairing in Example E above, combining two input-side transformations is order-dependent. This is real, checkable mathematics, not a citation from any past paper — no WMA11 series in the verified-facts corpus was found testing a doubly-combined trig transformation, so this section is presented as spec-required theory (verifiable directly), clearly separated from the citation-grounded findings in §5.

Compare $y = \sin(2x + \tfrac{\pi}{6})$ against $y = \sin\!\left(2\left(x+\tfrac{\pi}{6}\right)\right) = \sin\!\left(2x + \tfrac{\pi}{3}\right)$ . These look almost identical on the page — both start from $\sin 2x$ and both involve $\pi/6$ — but they are different functions: substituting $x=0.9$ gives $\sin(2(0.9)+\pi/6) \approx 0.730$ against $\sin(2(0.9+\pi/6)) \approx 0.290$ , a difference too large to be rounding. The first expression stretches $x$ by $2$ then adds a phase of $\pi/6$ directly to the (already-doubled) input — equivalent to translating $\sin 2x$ left by only $\pi/12$ , not $\pi/6$ , because the stretch compresses the translation too. The second expression translates $x$ left by $\pi/6$ first, then stretches — a translation of the full $\pi/6$ , applied before the compression touches it. Reading a transformation correctly off a combined equation requires identifying which operation the spec's own bracketing applies first, not assuming every $\pi/6$ -shaped number in the expression means "shift by $\pi/6$ ."

#5. Symmetry beyond odd/even (spec 3.3's third word, extended)

The global odd/even facts (§1, already fully taught by the live lesson) describe symmetry of the whole graph about one fixed point or line. Sine and cosine graphs also carry a repeating local symmetry, one instance per period, which is a distinct (if related) mathematical fact and is not stated anywhere in the existing WMA11 material on this topic:

$\sin(\pi - x) = \sin x$ — verified numerically. Geometrically: $y=\sin x$ has a vertical mirror line through every peak, e.g. at $x = \pi/2$ , reflecting the rising quarter-cycle before it onto the falling quarter-cycle after it.
$\cos(\pi - x) = -\cos x$ — verified numerically. Geometrically: $y = \cos x$ has point (rotational) symmetry about $(\pi/2, 0)$ — a zero of the function — not just about the single global point used for the odd/even classification.

Micro-example F: confirm the mirror line for sine at its first peak, $x=\pi/2$ : $\sin(\pi - \pi/3) = \sin(2\pi/3) = \sin(\pi/3)$ — both equal $\tfrac{\sqrt3}{2}$ , confirming the two points $x=\pi/3$ and $x=2\pi/3$ (equidistant from $\pi/2$ ) give the same height, exactly what a mirror line at $x=\pi/2$ predicts.

#6. Every real exam angle this exact spec point has actually been tested from

The facts bank's own frequency layer (signal-frequency-and-boundaries.md §1) reports "Trigonometry: seen in 4 of 5" fully-read series and "Graph transformations (spec 1.12): seen in 4 of 5" — both figures overstate this specific spec point (3.3, trig graphs) if read uncritically, because most of what feeds each count is a different spec item:

The Trigonometry-strand 4/5 figure is driven mainly by spec 3.1 (sine/cosine rule, ambiguous case — Jan 2019 Q7) and spec 3.2 (sector area/arc length — Jan 2023 Q6, Oct 2020 Q3, Oct 2021 Q7), none of which is spec 3.3.
The spec-1.12 "graph transformations" 4/5 figure is driven mainly by reciprocal-function transformations (Jan 2023 Q7(b), $y=6/x \to y=6/(x-2)$ ; Jan 2019 Q8, a reciprocal curve's asymptote and intersection-count question) — not trig functions.

Once those are excluded, the confirmed real exam angles on spec 3.3 specifically, across the nine series the facts bank has touched at all, are exactly three, spanning 3 of the 5 fully-read series:

Jan 2023 Q9(a) — "State the period of tan x," 1 mark (B1). Mark scheme, in full: "Period is π (radians) but condone 180° or just 180." Examiner report: fewer than half the cohort answered correctly; a recurring wrong answer was the interval $-\pi/2 < x < \pi/2$ , attributed by the report to a "confusion between period and domain/range." This is the single richest citation on the topic and is the one the live lesson's entire worked chain is built around.
Jan 2019 Q5 — sketch $y=\sin x$ on axes already drawn for $y=\cos 2x$ , no scale supplied. Examiner report: the missing scale on the given graph "lead to some candidates not sketching [ $\sin x$ ] carefully enough," and separately, many candidates left their own axis unscaled despite being prompted to mark one.
Oct 2021 Q4(a) — not a "describe the transformation" question, despite how the facts bank's own §4.1 entry frames it (a framing this document initially inherited and is correcting here, on the strength of two independent, already-corrected sibling documents in this same technique-hub — graph-transformations-in-words.ts's own codex deep-dive and SIGNAL brief, and this topic's own SIGNAL brief — all three of which re-checked the primary paper and found the same thing). The real question gives a sketch of $y=\cos 2x°$ over $0 \leqslant x \leqslant k$ , with a minimum point $Q$ and a point $R(k,0)$ marked, and asks candidates to state $Q$ 's coordinates and the value of $k$ : 3 marks, all B1, no method line — one B1 for either coordinate of $Q$ correct in position, a second B1 for both fully correct, a third B1 for $k=225$ (the sketch shows one-and-a-quarter periods of a $180°$ -period curve, so this mark is itself a period-reading mark). The wrong-answer commentary the facts bank preserves — candidates "mistakenly thinking" the transformation mapping $\cos x$ to $\cos 2x$ is a stretch in the $y$ -axis direction rather than the $x$ -axis — is real and belongs here, but it explains why some candidates mis-plotted $Q$ or mis-stated $k$ on this coordinates-and-period question, not a separate worded-description sub-part that does not exist on the real paper.

Explicitly excluded, for scope, not overlooked: Jan 2019 Q8 (reciprocal-function reflection/intersection-count question — confirmed by the facts bank's own lesson-audit log to belong to graph-transformations-in-words.ts's spec 1.12/reciprocals scope, not this one); Oct 2020 Q2/Q5 and Jan 2023 Q7(b) (both reciprocal-function transformations, same reason).

What this means for exam weight: on the one series where a full mark-by-mark breakdown exists (Jan 2023), spec-3.3 content contributed at most the 1 mark of Q9(a) directly (Q9(b)'s other 3 marks are a root-counting skill under spec 1.11, not 3.3) — the smallest confirmed single-topic mark contribution anywhere in this course's WMA11 mark-weight table. That is consistent with three confirmed sightings in nine series checked, not with three sightings being an exhaustive count of how often spec 3.3 can appear — spec 3.3 is compulsory content and a thin research trail is a fact about which series were read in depth, not a ceiling on the spec.

#7. Flagged gaps — what this document does not, and cannot yet, claim

No June-series paper has been read for spec-3.3 content specifically (the facts bank's two June-series reads were targeted at spec 1.3 and 1.6 questions only) — a real gap in the underlying evidence base, inherited here, not created by this document.
Oct 2022 and Jan 2024 are confirmed-genuine mark schemes in the facts bank but were never mined question-by-question, so either could contain an unread spec-3.3 question.
§4 and the reflection sub-case in §3 (Micro-example D) are stated as general, independently-verified mathematics required by spec 1.12/3.3's own wording, not as citation-grounded exam findings — no WMA11 series in the facts bank was found testing either a reflected trig graph or a doubly-combined input-side transformation, and this document does not claim otherwise.
Grade boundaries are not stated anywhere in this document, consistent with every other WMA11 technique-hub document in this course: the one figure the facts bank has (October 2022, from an unverified web search) is explicitly marked UNVERIFIED there and is not repeated here.
No likelihood percentage or "expect this on your paper" claim is made anywhere above — every frequency statement is "seen in N of M series checked," per this course's standing rule.