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TMUA Paper 1 Deep Dive: Trigonometry & Logarithms
30th September, 2026 | Abhishek Malani | 14 Min Read
Introduction
Trigonometry and logarithms are two of the highest-yield topics for TMUA Paper 1 practice, precisely because they are compact, rule-based areas where a small number of identities and laws generate a large share of the questions. This deep dive works through the official specification content for TMUA trigonometry and TMUA logarithms, sets out the TMUA trig identities and TMUA log laws you are actually expected to know from memory, since there is no formula booklet, and finishes with original practice questions and worked solutions in the TMUA style so you can test yourself before exam day.
Table of Contents
Why Trig and Logs Matter So Much on Paper 1
- No formula booklet is provided: The official specification states explicitly that candidates are expected to understand and recall all relevant formulae; every identity and law covered here needs to be memorised, not looked up.
- No calculator is allowed: Every trig and log question is designed to be solvable using exact values, algebraic manipulation, and reasoning, not decimal approximation, so fluency with exact values and rule manipulation matters more than raw computation.
- These topics reward pattern recognition over raw content volume: Unlike, say, coordinate geometry or sequences, trig and logs are governed by a short, fixed list of rules; once those rules are second nature, a large share of related questions become fast, reliable marks.
- They frequently combine with other topics: Trig equations often sit inside graph-transformation or equation-counting questions, and log laws routinely appear inside exponential-equation and sequence questions, so weakness here quietly costs marks elsewhere too.
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TMUA Trig Identities: What You Must Know Cold
The official TMUA specification sets out a focused, specific list of trigonometry content (references TR1–TR6). This is the exact scope, no more and no less:
| Spec Ref | What It Covers |
|---|---|
|
TR1 |
The sine rule and cosine rule, and the area of a triangle as ½ab sin C. Includes understanding the “ambiguous case” (angle-side-side) for the sine rule. Problems may be set in 2D or 3D.
|
|
TR2 |
Radian measure, including its use for arc length (rθ) and the area of a sector and segment.
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TR3 |
Exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, 90°, known without a calculator.
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TR4 |
The sine, cosine, and tangent functions: their graphs, symmetries, and periodicity.
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TR5 |
Knowledge and use of tanθ = sinθ/cosθ and sin²θ + cos²θ = 1.
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TR6 |
Solving trigonometric equations in a given interval, which may require the TR5 identities.
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Two identities in particular, tanθ = sinθ/cosθ and sin²θ + cos²θ = 1, are the workhorses of almost every trig question beyond the most basic exact-value lookup. Being able to rearrange sin²θ + cos²θ = 1 instantly into cos²θ = 1 − sin²θ or sin²θ = 1 − cos²θ without pausing to think is what separates a fast, confident answer from a stalled one under time pressure.
TMUA Paper 1 Trigonometry: The Core Skills
Translating the specification into practical exam skills, here is what TMUA Paper 1 trigonometry questions actually ask you to do:
- Recall exact values instantly: sin, cos, and tan of 0°, 30°, 45°, 60°, and 90° (and their radian equivalents: 0, π/6, π/4, π/3, π/2) need to be automatic, since no calculator is available and these values underpin equation-solving questions.
- Convert fluently between degrees and radians, and apply radian measure directly in arc length (s = rθ) and sector/segment area formulae without needing to convert back to degrees first.
- Use the sine and cosine rules confidently, including the ambiguous case: TMUA questions test whether you recognise when the sine rule gives two possible triangles (angle-side-side) rather than one, which is a common source of lost marks.
- Substitute the sin²θ + cos²θ = 1 identity to turn a mixed-function equation into a single-function quadratic: This is the single most common trig technique tested, converting something like 12cos²θ + 6sinθ − 10 = 2 into a solvable quadratic in sinθ.
- Count solutions systematically within a given interval: Many TMUA trig questions ask not for the answer itself but for how many solutions exist in a stated range, which rewards a clear, systematic approach (sketching the graph or tracking each branch) over guesswork.
- Read graph symmetry and periodicity to solve equations without full algebra: Recognising that sin and cos repeat every 2π (or 360°), and that their graphs are reflections/shifts of one another, often shortcuts a question that looks algebraically heavy.
TMUA Logarithms: The Full Rule Set
The specification’s exponentials and logarithms content (references EL1–EL3) is short but dense, and nearly every question in this area draws on it directly:
| Spec Ref | What It Covers |
|---|---|
|
EL1 |
y = a^x and its graph, for simple positive values of a.
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EL2 |
The laws of logarithms: a^b = c ⇔ b = logₐc; logₐx + logₐy = logₐ(xy); logₐx − logₐy = logₐ(x/y); k·logₐx = logₐ(x^k); plus the special cases logₐ(1/x) = −logₐx and logₐa = 1. Note: the change of base formula is explicitly excluded from TMUA questions.
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EL3 |
Solving equations of the form a^x = b, and equations reducible to that form after algebraic manipulation, such as 3^(2x) = 4, or quadratic-in-disguise equations like 2^(5x) − 3×5^x + 2 = 0.
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The exclusion of the change-of-base formula in EL2 is worth underlining: the TMUA will not ask you to convert between log bases using that rule, which narrows what you actually need to memorise compared with a full A-level logs syllabus.
TMUA Log Laws: Common Traps
Most TMUA log laws errors come from a small set of repeated mistakes, all worth drilling out of your technique before the exam:
- Confusing logₐx + logₐy = logₐ(xy) with logₐx × logₐy. The addition law combines logs into a single log of a product; it does not multiply the log values together. This mix-up is one of the most common errors on log questions generally.
- Forgetting that k·logₐx = logₐ(x^k) works in both directions. Candidates often use this law to simplify but forget it can also expand a log of a power back into a coefficient, which is frequently the key step in solving an equation.
- Missing domain restrictions. logₐx is only defined for x > 0, so an algebraic solution that gives a negative or zero value inside a log must be rejected, a check that is easy to skip under time pressure but is sometimes exactly what a question is testing.
- Not recognising a disguised quadratic. Equations like a^(2x) − k×a^x + m = 0 are quadratics in a^x; substituting u = a^x turns them into a standard quadratic, but many candidates try to solve them directly with log laws first and get stuck.
- Misapplying the special cases. logₐ(1/x) = −logₐx and logₐa = 1 are exact, memorisable shortcuts; working them out from first principles every time wastes valuable seconds on a non-calculator paper.
Worked Practice Question: Trigonometry
Solve 2sin²θ + sinθ − 1 = 0 for 0° ≤ θ < 360°. How many solutions does the equation have in this range?
- A) 1
- B) 2
- C) 3
- D) 4
Worked solution: Treat this as a quadratic in sinθ. Let s = sinθ, so 2s² + s − 1 = 0. Factorising: (2s − 1)(s + 1) = 0, giving s = 1/2 or s = −1.
For sinθ = 1/2: using the TR3 exact values, θ = 30° or θ = 150° (the second solution comes from the symmetry of the sine graph about 90°), both within the given range. That is two solutions so far.
For sinθ = −1: this is the single minimum point of the sine graph, occurring once per period at θ = 270°. That is one further solution.
Total: 30°, 150°, and 270°, which is three solutions. The answer is C) 3.
This question rewards exactly the technique flagged above: recognising the disguised quadratic in sinθ, applying exact-value recall, and using graph symmetry to find the second solution rather than trying to derive it algebraically.
Worked Practice Question: Logarithms
Solve log₂(x + 3) + log₂(x − 1) = 5.
- A) x = 5
- B) x = 6
- C) x = 29
- D) x = 13
Worked solution: Apply the addition law, logₐx + logₐy = logₐ(xy): log₂[(x + 3)(x − 1)] = 5.
Convert from log form using a^b = c ⇔ b = logₐc: (x + 3)(x − 1) = 2⁵ = 32.
Expand: x² + 2x − 3 = 32, so x² + 2x − 35 = 0. Factorising: (x + 7)(x − 5) = 0, giving x = −7 or x = 5
Now apply the domain check flagged above: both (x + 3) and (x − 1) must be strictly positive for the original logs to be defined. For x = −7, (x − 1) = −88, which is negative, so this solution is rejected. For x = 5, (x + 3) = 8 and (x − 1) = 4, both positive, so this solution is valid.
The answer is A) x = 5. This question is a direct test of two things covered above: correctly applying the addition law in the right direction, and remembering the domain restriction that catches out anyone who stops at the quadratic’s two roots without checking them.
Worked Practice Question: Trig and Logs Combined
This style of question, combining two topics in one problem, is common on Paper 1 and rewards exactly the kind of fluency this deep dive is built around. Given that sinθ = 3^(−x) for some real x, and that sinθ = 1/3 when θ is a value satisfying 0 < θ < π/2, find x.
- A) x = log₃(3)
- B) x = 1
- C) x = 3
- D) x = log₃(9)
Worked solution: Since sinθ = 1/3, and sinθ = 3^(−x), set 3^(−x) = 1/3.
Rewrite 1/3 as 3⁻¹ (using the special case logₐ(1/x) = −logₐx in reverse, or simply index laws): 3^(−x) = 3⁻¹.
Since the bases match, equate the exponents: −x = −1, so x = 1. The answer is B) x = 1.
The trap here is treating this as a full logarithm equation requiring EL2’s laws in their expanded form, when recognising both sides as powers of the same base (an EL1/EL3-style skill) solves it in one line. Spotting the faster route is exactly the kind of judgement Paper 1 rewards under a 75-minute, 20-question time limit.
A Quick Self-Test Checklist
- Can you state sin, cos, and tan of 0°, 30°, 45°, 60°, and 90° without hesitating?
Can you convert 12cos²θ + 6sinθ − 10 = 2 into a quadratic in sinθ in under 30 seconds? - Do you automatically check whether a sine-rule question is the ambiguous (angle-side-side) case?
- Can you state all four log laws from EL2, including both special cases, from memory?
- Do you automatically substitute u = a^x when you see an equation of the form a^(2x) − k·a^x + m = 0?
- Do you check the domain (x > 0) every time you solve a log equation, before finalising an answer?
If any of these feel slow or uncertain, that is exactly where focused revision time before the test will pay off fastest, since these are high-frequency, rule-based skills rather than topics requiring broad content coverage.
Summary
TMUA trigonometry and TMUA logarithms are compact but heavily tested areas of Paper 1, governed by a short, fixed set of rules laid out precisely in references TR1–TR6 and EL1–EL3 of the official specification. The TMUA trig identities to know cold are tanθ = sinθ/cosθ and sin²θ + cos²θ = 1, alongside exact values at the five key angles and fluent use of the sine and cosine rules, including the ambiguous case. The TMUA log laws center on the addition, subtraction, and power rules plus two special cases, explicitly excluding the change-of-base formula, and the most common errors come from misapplying these laws or forgetting the x > 0 domain restriction. Because there is no formula booklet and no calculator, real fluency, not just familiarity, with both topics is what converts into fast, reliable marks on TMUA Paper 1 practice questions, especially where trig and logs are combined with other topics in a single question.
STEP and TMUA are mathematics admissions tests used by UK universities for different courses and at different stages of the admissions process. STEP consists of extended mathematical problem-solving papers and is used extensively by Cambridge, particularly for Mathematics. TMUA is a computer-based test assessing mathematical thinking and reasoning and is used by Cambridge and a number of other universities for selected courses. Students should first check the requirements for their specific university, course and year of entry, and then plan their preparation around the relevant assessment.
FAQs
The specification requires tanθ = sinθ/cosθ and sin²θ + cos²θ = 1 (reference TR5), plus exact values of sine, cosine, and tangent at 0°, 30°, 45°, 60°, and 90° (TR3), and fluent use of the sine rule, cosine rule, and triangle area formula (TR1). There is no formula booklet, so all of this must be recalled from memory.
No. The official specification explicitly states that questions requiring knowledge of the change of base formula will not be set, which narrows the log laws you need to memorise to the addition, subtraction, and power laws plus the two listed special cases.
Convert the equation into a standard form (often a quadratic in sin, cos, or tan), find all values of the trig function that satisfy it, then use exact values and graph symmetry/periodicity to find every matching angle within the stated interval, checking systematically rather than guessing.
Confusing the addition law, logₐx + logₐy = logₐ(xy), with multiplying the log values themselves, and forgetting to check that arguments of a log remain positive (x > 0) after solving an equation, which can invalidate an otherwise correct algebraic solution.
No. The TMUA is entirely non-calculator, so trig and log questions are designed to be solved using exact values, algebraic manipulation, and index/log laws rather than decimal computation.
Want more TMUA Paper 1 practice questions with worked solutions, tailored to your weak topics? Reach out to Quest for Success; we’re here to help you build genuine fluency before test day.
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