A Level · Cambridge (CIE) · 9709

Sketch secant cosecant and cotangent with separate branches

Use the zeros and signs of the original ratios to locate asymptotes, turning points, periods and ranges.

Pure Mathematics 2 & 3 · Trigonometry · 2.3 / 3.3

Exam scope and assessment notes

Pure Mathematics 2 (Paper 2, AS only) and Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027. Paper 1 knowledge is assumed. The AS route using Papers 1 + 2 cannot be carried forward to A Level. Shared sections 2.3 and 3.3 cover all six trigonometric functions, reciprocal squared identities, compound and double angles, and amplitude-phase forms used in simplification, exact evaluation and equations. State angle units, keep original domains and find every solution in the supplied interval. Written practice is self-checked.

Before you begin

Sketch sine, cosine and tangent and use their zeros.
Evaluate reciprocal ratios and recognise an asymptote.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which range belongs to sec x?

What you will learn

Sketch the reciprocal functions over a stated radian interval without joining excluded inputs.
Compare the domains, ranges and periods of all six trigonometric functions.

Secant, y = 1/cos x, has period 2π and range y ≤ −1 or y ≥ 1. Its excluded vertical asymptotes in 0 ≤ x ≤ 2π are π/2 and 3π/2. The marked extrema are (0, 1), (π, −1) and (2π, 1). The separate branches are clipped at y = ±3 for this drawing and continue unbounded towards the dashed asymptotes; the clipping edges are not function endpoints. Secant has no zeros.

Cosecant, y = 1/sin x, has period 2π and range y ≤ −1 or y ≥ 1. Its excluded vertical asymptotes in 0 ≤ x ≤ 2π are 0, π and 2π. The positive branch has minimum (π/2, 1); the negative branch has maximum (3π/2, −1). The separate branches are clipped at y = ±3 and continue unbounded towards the dashed asymptotes. Neither the asymptote inputs nor a zero output are permitted.

Cotangent, y = cos x/sin x, has period π and range all real numbers. Its excluded vertical asymptotes in 0 ≤ x ≤ 2π are 0, π and 2π. Each separate branch decreases through a marked zero at π/2 or 3π/2. These zeros are valid cotangent inputs although tangent is undefined there. The branches continue unbounded beyond the y = ±3 clipping edges; never join across a dashed asymptote.

Build reciprocal graphs from their denominators

Sine and cosine have all-real input domains, range [−1, 1] and period 2π. Their reciprocal secant and cosecant both have range y ≤ −1 or y ≥ 1 and period 2π. Neither reciprocal can be zero. Secant is undefined where cosine vanishes: on 0 ≤ x ≤ 2π these inputs are π/2 and 3π/2. Its labelled extrema are (0, 1), (π, −1) and (2π, 1), with separate branches tending to vertical asymptotes.

Cosecant is undefined where sine vanishes: x = 0, π, 2π in this interval. On (0, π) it is positive with minimum (π/2, 1); on (π, 2π) it is negative with maximum (3π/2, −1). Near a denominator zero, the magnitude grows without bound. The endpoint asymptotes remain excluded even when the drawing window includes 0 and 2π. A graph clipped to a finite vertical window has continuing branches, not extra endpoints.

Cotangent changes the zero and asymptote positions

Tangent and cotangent have range all real numbers and period π. Tangent is undefined where cosine is zero and has zeros where sine is zero. Cotangent reverses those roles: its denominator sine excludes 0, π and 2π here, while its zeros occur at π/2 and 3π/2. Each cotangent branch decreases from positive unbounded values to negative unbounded values as x runs across (0, π) or (π, 2π). It is not a tangent graph with unchanged asymptotes.

For y = sec 2x, the factor 2 halves the period to π. On 0 ≤ x ≤ π, the asymptotes are π/4 and 3π/4, and key points are (0, 1), (π/2, −1) and (π, 1). Its range still excludes values strictly between −1 and 1. Use the transformed denominator cos 2x to locate forbidden inputs before solving sec 2x = 2 or connecting any branches.

Pause and explain

Which statement correctly describes cot x on 0 < x < π?

Put the idea to work

Worked example

Sketch y = sec x for 0 ≤ x ≤ 2π, giving its asymptotes, key extrema, period and range.

Show the worked solution

The zeros of cos x locate excluded asymptotes at π/2 and 3π/2.
Reciprocate cos values 1, −1, 1 at 0, π, 2π to get the three key extrema (0, 1), (π, −1), (2π, 1).
Draw separate signed branches approaching the asymptotes; the period is 2π and range is y ≤ −1 or y ≥ 1.

Answer Asymptotes π/2, 3π/2; extrema (0, 1), (π, −1), (2π, 1); period 2π; range y ≤ −1 or y ≥ 1.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Sketch cosec x for 0 ≤ x ≤ 2π, stating the asymptotes, signed extrema and period.

Give me a hint

Use sine's zeros for the excluded inputs and its ±1 values for the reciprocal extrema.

Compare my reasoning

Sine vanishes at 0, π and 2π, so these are excluded vertical asymptotes.
The positive branch has minimum (π/2, 1); the negative branch has maximum (3π/2, −1).
The period is 2π, with range y ≤ −1 or y ≥ 1; neither branch crosses zero.

Asymptotes 0, π, 2π; minimum (π/2, 1), maximum (3π/2, −1); period 2π; range y ≤ −1 or y ≥ 1.

Look for these in your work

Locate asymptotes from the original denominator.
Keep signs and branch extrema distinct.

2 · Independent

Sketch cot x over 0 ≤ x ≤ 2π and compare its period and asymptotes with tan x.

Give me a hint

Use cos x/sin x, so sine controls forbidden inputs and cosine controls the zeros.

Compare my reasoning

Cotangent excludes 0, π and 2π and has zeros at π/2 and 3π/2.
Draw a decreasing branch on each interval (0, π) and (π, 2π), with range ℝ and period π.
Tangent also has period π and range ℝ, but its asymptotes are π/2 and 3π/2 here and its branches increase.

cot: asymptotes 0, π, 2π; zeros π/2, 3π/2; decreasing branches; range ℝ; period π. Tangent's asymptotes are π/2, 3π/2.

Look for these in your work

Use the correct denominator rather than copy tangent's graph.
State separate branches, range and period.

3 · Transfer

For y = sec 2x on 0 ≤ x ≤ π, state the asymptotes and solve y = 2 exactly.

Give me a hint

First find the denominator zeros, then convert the valid reciprocal equation to cos 2x = 1/2.

Compare my reasoning

cos 2x = 0 at x = π/4 and 3π/4, giving the two excluded asymptotes.
The equation sec 2x = 2 gives cos 2x = 1/2; 0 ≤ 2x ≤ 2π supplies angles π/3 and 5π/3.
Divide by 2 to obtain x = π/6 and 5π/6; both lie in the interval and have non-zero cosine denominator.

Asymptotes x = π/4, 3π/4; y = 2 at x = π/6 or 5π/6.

Look for these in your work

Apply the input scaling to both asymptotes and equation angles.
Check the original reciprocal domain for every solution.

Common mistakes

Secant and cosecant have no zeros and no finite amplitude.
Mark cotangent's sine zeros as asymptotes rather than use tangent's cosine zeros.
An excluded asymptote is not an attained endpoint; separate branches must stay separate.

Recall without your notes

What can you explain now?

What are cotangent's zeros and excluded asymptotes in 0 ≤ x ≤ 2π?

Compare with the explanation

Zeros π/2 and 3π/2; excluded asymptotes 0, π and 2π.

Cosine zeros give cotangent zeros when sine is non-zero; sine zeros make its defining quotient undefined.

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Make it stick

Sketch all six functions for one repeated cycle. Explain why the reciprocal functions' zeros, ranges and excluded inputs differ from their original ratios.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.