IB Maths AA HL Topic 3 — Geometry & Trigonometry Paper 1 & 2 ~6 min read

Exact Values

For certain “nice” angles — 0°, 30°, 45°, 60°, 90° and their multiples — sin, cos, and tan can be written exactly using fractions and surds, instead of decimal approximations. Memorise the small table once, and use unit-circle symmetries to handle every other multiple.

📘 What you need to know

Memorise exact values for 0°, 30°, 45°, 60°, 90°, 180° (and 0, π/6, π/4, π/3, π/2, π in radians).
Two source triangles: a 30°-60°-90° (from a split equilateral) and a 45°-45°-90° (right isosceles).
For other multiples of 30° or 45°, use the unit-circle symmetries from the previous note: 180° − θ, 180° + θ, 360° − θ.
tan 90° is undefined (vertical line — division by 0).
Exact form = surds and fractions; calculator decimals lose precision.
Sketch the triangles in your exam if you forget — re-derive in 30 seconds rather than guess.

The exact values table

Angle	0°	30° (π/6)	45° (π/4)	60° (π/3)	90° (π/2)	180° (π)
sin	0	1/2	1/√2	√3/2	1	0
cos	1	√3/2	1/√2	1/2	0	−1
tan	0	1/√3	1	√3	undefined	0

Quick pattern: sin goes 0, 1/2, 1/√2, √3/2, 1 — increasing. Cos is the reverse. Tan = sin/cos.

The two special triangles

30°-60°-90°

sides 1, √3, 2

half of an equilateral triangle (side 2)

45°-45°-90°

sides 1, 1, √2

right-isosceles triangle (legs 1)

From the 30-60-90: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3. From the 45-45-90: sin 45° = cos 45° = 1/√2, tan 45° = 1.

Extending to other multiples

For multiples of 30° or 45° outside the table, write the angle as 180° ± θ or 360° − θ for an acute θ, then apply the symmetry rule and the CAST quadrant signs.

Examples sin 150° = sin(180° − 30°) = +sin 30° = 1/2
cos 210° = cos(180° + 30°) = −cos 30° = −√3/2
tan 315° = tan(360° − 45°) = −tan 45° = −1

🧭 Recipe — finding exact trig values

Reduce the angle to within 0° – 360° by adding/subtracting 360°.
Identify the quadrant using CAST. This tells you the sign.
Find the related acute angle: 180° − θ in Q2, θ − 180° in Q3, 360° − θ in Q4.
Look up the acute trig value from the table (or derive from a sketch).
Apply the sign from CAST. Done.

Worked examples

WE 1

Direct lookup from the table

Find the exact value of sin 60° + cos 30°.

Step 1: Read both values from the table sin 60° = √3/2 cos 30° = √3/2 Step 2: Add √3/2 + √3/2 = 2(√3/2) = √3 sin 60° + cos 30° = √3 sin 60° = cos 30° because sin θ = cos(90° − θ) — co-function identity

WE 2

Exact values for an angle in Q2

Find the exact values of sin 150° and cos 150°.

Step 1: Identify the quadrant 150° is in Q2 (90° < 150° < 180°) CAST: only sin is positive Step 2: Write 150° = 180° − 30° and apply symmetries sin 150° = sin(180° − 30°) = +sin 30° = 1/2 cos 150° = cos(180° − 30°) = −cos 30° = −√3/2 sin 150° = 1/2; cos 150° = −√3/2

WE 3

Exact value of tan in Q3

Find the exact value of tan 225°.

Step 1: Identify the quadrant 225° is in Q3 (180° < 225° < 270°) CAST: tan is positive in Q3 Step 2: Write 225° = 180° + 45° and apply symmetry tan 225° = tan(180° + 45°) = +tan 45° tan 45° = 1 tan 225° = 1 in Q3 both sin and cos are negative, so tan = sin/cos = (negative)/(negative) = positive

WE 4

Exact value in Q4

Find the exact value of cos 330°.

Step 1: Identify the quadrant 330° is in Q4 (270° < 330° < 360°) CAST: cos is positive in Q4 Step 2: Write 330° = 360° − 30° and apply symmetry cos 330° = cos(360° − 30°) = +cos 30° = √3/2 cos 330° = √3/2

WE 5

Exact value in radians — combined sum

Find the exact value of sin(5π/4) + cos(5π/4).

Step 1: Convert to degrees (or work in radians directly) 5π/4 = 225° → Q3 CAST: only tan is positive in Q3 → sin and cos both negative Step 2: Use the symmetry 5π/4 = π + π/4 sin(5π/4) = −sin(π/4) = −1/√2 cos(5π/4) = −cos(π/4) = −1/√2 Step 3: Add −1/√2 + (−1/√2) = −2/√2 = −√2 sin(5π/4) + cos(5π/4) = −√2 2/√2 simplifies: multiply numerator and denominator by √2 → 2√2/2 = √2

WE 6

Derive exact values from a 45°-45°-90° triangle

Using a right-isosceles triangle with legs of length 1, derive the exact values of sin 45°, cos 45°, and tan 45°.

Step 1: Sketch — right triangle with two legs = 1 and the right angle between them the other two angles are equal → both = 45° Step 2: Apply Pythagoras for the hypotenuse hyp² = 1² + 1² = 2 hyp = √2 Step 3: Apply SOH CAH TOA for the 45° angle sin 45° = O/H = 1/√2 cos 45° = A/H = 1/√2 tan 45° = O/A = 1/1 = 1 sin 45° = cos 45° = 1/√2; tan 45° = 1 1/√2 can also be written √2/2 (rationalised denominator) — both are correct exact forms

💡 Top tips

Sketch the two triangles at the start of every Paper 1 question. Re-derive sin, cos, tan in 30 seconds rather than guessing.
CAST tells the sign; the table tells the magnitude. Together they handle any angle.
1/√2 = √2/2 — both are accepted. Most exam mark schemes prefer the rationalised form (√2/2) but won’t penalise either.
“Exact” means surds, not decimals. 1/2 ✓, 0.5 ✗ (in exact-form questions).
For radians, the same table applies: π/6, π/4, π/3, π/2, π, 2π. Convert mentally if needed.

⚠ Common mistakes

Confusing sin 30° and cos 30°. sin 30° = 1/2; cos 30° = √3/2. The “smaller” value belongs to the smaller angle for sin, larger angle for cos.
Wrong sign for the quadrant. CAST is non-negotiable — always check.
Writing decimals instead of exact form. 0.866 ≠ √3/2 in an exact-value question.
Forgetting tan 90° is undefined. Don’t write 1 or “infinity” — write “undefined” (or note that the cosine is 0).
Using the wrong triangle. 30-60-90 for 30° and 60°; 45-45-90 for 45°. Mixing them up gives wrong values.

That closes the unit-circle foundations. The next section moves into Trigonometric Functions & Graphs — plotting sin, cos, tan as functions of x, then transforming and modelling them. The exact values from this note will keep coming up — they’re the labelled points on every trig graph you’ll draw from now on.

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