IB DP Maths AA · Exam special · SL & HL · the no-calculator paper · ~8 min read
IB Maths AA Paper 1: Non-Calculator Skills and Exact Values
The skills IB Maths AA Paper 1 tests without a calculator: exact trig values and the unit circle, logs, surds and indices, arithmetic shortcuts, sketching by hand, solving trig equations and the slips examiners see every session.
Paper 1 formatExact trig valuesUnit circle & CASTIndices, logs & surdsFast arithmeticSketching by handTrig equationsCalculus without a GDC
RSby Riyaz Sir
mentorriyaz.in
mentorriyaz.in
No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againHANDY worth memorisingTest yourself at the end
In 30 seconds
- Paper 1 is the no-calculator paper in Maths AA: 80 marks in 1 hour 30 minutes at SL, 110 marks in 2 hours at HL. It is 40% of the SL grade and 30% of the HL grade.
- The numbers are chosen to work out nicely. If your arithmetic gives something ugly, such as √7.3 or 0.4286…, check your working.
- Answers must be exact: fractions, surds, π, e and ln, not decimals. Examiners repeatedly penalise unfinished forms like log₂16 or 0.2/0.35.
By the end you can
- Recall exact trig values and the unit circle instantly.
- Use index, log and surd laws fluently without a calculator.
- Sketch graphs and solve equations by hand.
- Present answers in their simplest exact form.
1The paper at a glance
| SL | HL | |
|---|---|---|
| Marks | 80 | 110 |
| Time | 1 h 30 min | 2 h |
| Weighting | 40% | 30% |
| A 7 needed (May 2025) | 50–60 out of 80 | 70–76 out of 110 |
| A 4 needed (May 2025) | 18–24 out of 80 | 26–31 out of 110 |
- Section A: short questions (4–8 marks). Section B: a few long questions (14–20 marks each). Section B parts are linked; later parts often use earlier results.
- 5 minutes of reading time: spot the easy marks and the questions with "exact" in them.
2Exact trig values
The table to know by heart
| θ (degrees) | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| θ (radians) | 0 | π/6 | π/4 | π/3 | π/2 |
| sin θ | 0 | ½ | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | ½ | 0 |
| tan θ | 0 | √3/3 (= 1/√3) | 1 | √3 | undefined |
- Memory trick: sin of 0°, 30°, 45°, 60°, 90° is √0/2, √1/2, √2/2, √3/2, √4/2. Cosine is the same list backwards.
- Not in the formula booklet. Neither is the conversion 180° = π radians.
The unit circle and CAST
- Signs by quadrant (anticlockwise from the positive x-axis): All positive in the 1st, Sin in the 2nd, Tan in the 3rd, Cos in the 4th.
- Related angles: sin(π − θ) = sin θ; cos(π − θ) = −cos θ; sin(π + θ) = −sin θ; cos(2π − θ) = cos θ; sin(−θ) = −sin θ; cos(−θ) = cos θ.
Worked example
Find the exact value of cos(5π/6) and of sin(4π/3).
- 5π/6 = π − π/6 is in the 2nd quadrant, where cos < 0: cos(5π/6) = −√3/2
- 4π/3 = π + π/3 is in the 3rd quadrant, where sin < 0: sin(4π/3) = −√3/2
Worked example
Given sin θ = 3/5 and π/2 < θ < π, find cos θ and sin 2θ exactly.
- cos²θ = 1 − 9/25 = 16/25, and cos θ < 0 in the 2nd quadrant, so cos θ = −4/5
- sin 2θ = 2 sin θ cos θ = 2 × 3/5 × (−4/5) = −24/25
Common mistakeForgetting the sign. "cos θ = ±4/5" or "4/5" both lose the mark when the quadrant is given.
3Trig equations by hand
- Find the basic angle, then every solution in the interval. Adjust the interval first if the angle is 2x or x + π/3.
- Don't divide by cos x or sin x: you lose solutions. Factorise instead.
Worked example
Solve 2 sin²x = sin x for 0 ≤ x ≤ 2π.
- 2 sin²x − sin x = 0 ⇒ sin x(2 sin x − 1) = 0 M1
- sin x = 0 ⇒ x = 0, π, 2π A1
- sin x = ½ ⇒ x = π/6, 5π/6 A1
- x = 0, π/6, 5π/6, π, 2π
Common mistakeDividing by sin x first and getting only π/6 and 5π/6. Three solutions lost.
4Indices, logs and surds
- Index laws: am × an = am+n; (am)n = amn; a−n = 1/an; a1/n = ⁿ√a; a⁰ = 1.
- Logs (laws are in the booklet; using them fast isn't): log₂32 = 5; log₃(1/9) = −2; ln e⁴ = 4; eln 7 = 7.
- Common errors: log(a + b) is not log a + log b; log 2³ = 3 log 2, not (log 2)³.
- Surds: √50 = 5√2; (√3 + 1)(√3 − 1) = 2. Rationalise with the conjugate: 4/(√5 − 1) = 4(√5 + 1)/4 = √5 + 1.
Worked example
Solve 4x = 8x − 1 exactly.
- Write both as powers of 2: 22x = 23(x − 1) M1
- 2x = 3x − 3 ⇒ x = 3 A1
Worked example
Solve log₂x + log₂(x − 2) = 3.
- log₂(x(x − 2)) = 3 ⇒ x² − 2x = 8 [M1A1]
- (x − 4)(x + 2) = 0 ⇒ x = 4 or x = −2 A1
- x − 2 > 0, so reject −2: x = 4 R1
5Arithmetic without a calculator
- Fractions before decimals: 0.2/0.35 = 20/35 = 4/7. Leave π, e and surds in the answer.
- Cancel before multiplying: (12 × 35)/(14 × 15) = 2.
- Squares to know: 11² to 20² (121, 144, 169, 196, 225, 256, 289, 324, 361, 400), and 25² = 625.
- Powers: 2¹⁰ = 1024, 3⁵ = 243, 5⁴ = 625.
- Binomial coefficients from Pascal's triangle: row 5 is 1, 5, 10, 10, 5, 1; row 6 is 1, 6, 15, 20, 15, 6, 1.
- Check each line. Most lost Paper 1 marks are sign or bracket slips: (−2)² = 4, (2x)³ = 8x³, −(x − 3) = −x + 3.
6Sketching by hand
- Mark the key points with exact values: intercepts, turning points, asymptotes (as equations, e.g. y = 3/2), endpoints of the domain.
- Quadratics: vertex form a(x − h)² + k gives the vertex (h, k) directly.
- Rational functions: vertical asymptote where the denominator is 0; horizontal asymptote from the leading coefficients.
- ǀf(x)ǀ: reflect the parts below the x-axis upwards. f(ǀxǀ): reflect the right-hand part in the y-axis.
7Calculus without a GDC
- Rewrite first: 5/x² = 5x⁻²; √x = x1/2.
- Remember the inside derivative: ∫e2x dx = ½e2x + C; ∫1/(2x + 1) dx = ½ ln ǀ2x + 1ǀ + C.
- Definite integrals in exact form: ∫₀π/2 cos x dx = sin(π/2) − sin 0 = 1.
- Tangent and normal lines: find the gradient by substituting into f'(x), then y − y₁ = m(x − x₁). The normal gradient is −1/m.
Common mistakeLeaving answers like ln 1 or e⁰ unsimplified: they are 0 and 1.
8Final form of answers
- Simplify to integers when possible: write 3, not √9 or 6/2.
- Fractions don't need to be in lowest terms, but the top and bottom must be integers.
- "Exact" means no decimals. 1.414 is not an exact form of √2.
- Angles in radians unless the question uses degrees.
Linking questionRevise the content behind these skills in Trig identities, equations and graphs, Exponents and logarithms and Integration and kinematics. For what is and isn't given in the exam, see the AA formula booklet guide.
Test yourself — cover the answers first
Quick check
- What is the exact value of tan(π/3)?
- What is sin(7π/6)?
- Simplify 3/(√3).
- Solve 9x = 27.
- Why can't you divide sin x cos x = sin x by sin x?
- Simplify log₄8.
Show answers
- √3
- −½
- √3
- x = 3/2
- you lose the solutions where sin x = 0
- 3/2
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