IB Maths AA SLTopic 3 — Geometry & TrigPaper 1 (no GDC)~9 min read
Exact Values
For a handful of “special” angles — 0°, 30°, 45°, 60°, 90° (and their multiples) — the values of sin, cos, and tan can be written exactly using simple fractions and surds. These come up constantly on Paper 1 (no GDC), so memorising them earns easy marks.
📘 What you need to know
Exact values are fractions and surds (no decimals) for sin, cos, and tan of special angles.
You need to know them for 0°, 30°, 45°, 60°, 90°, 180° (and 360°), in both degrees and radians.
In radians these are 0, π6, π4, π3, π2, π.
For any multiple of 30° or 45°, use the unit circle’s symmetries together with the basic exact values.
The values come from two special right-angled triangles: the 30°-60°-90° (from a halved equilateral triangle) and the 45°-45°-90° (an isosceles right triangle).
The exact values table
This is the table to memorise. It’s worth getting it tattooed on the inside of your eyelids — it shows up almost every Paper 1.
Degrees
0°
30°
45°
60°
90°
180°
360°
Radians
0
π6
π4
π3
π2
π
2π
sin
0
12
√22
√32
1
0
0
cos
1
√32
√22
12
0
−1
1
tan
0
1√3
1
√3
undef.
0
0
Why is tan 90° undefined? Because tan θ = sin θ / cos θ, and at 90° you have cos 90° = 0. Dividing by zero is undefined — so tan 90° has no value.
🧠 Memory trick — the sine pattern
For 0°, 30°, 45°, 60°, 90°, the sine values follow a beautiful pattern: √02, √12, √22, √32, √42 — that’s 0, ½, √2/2, √3/2, 1. Cosine is the same pattern in reverse. Once you spot this, the table is much easier to remember.
Where do these values come from? — the two special triangles
You don’t have to memorise the table blindly. The exact values come from two very simple triangles you can sketch in seconds. Once you’ve drawn them, just apply SOH CAH TOA.
🔺 The 30°-60°-90° triangle
How to derive it: start with an equilateral triangle, side length 2. Drop a perpendicular from one vertex — it splits the triangle into two identical 30°-60°-90° triangles, each with sides 1, √3, 2.
sin 30°12
cos 30°√32
tan 30°1√3
sin 60°√32
cos 60°12
tan 60°√3
🔺 The 45°-45°-90° triangle
How to derive it: start with an isosceles right-angled triangle where the two short sides each have length 1. By Pythagoras, the hypotenuse is √2. Both non-right angles must be 45°.
sin 45°1√2
cos 45°1√2
tan 45°1
🤔 Why “√2/2″ instead of “1/√2“?
Both are mathematically the same value, but in IB you usually want the denominator rationalised (no surds on the bottom). Multiply top and bottom by √2: 1√2 = √22. Same for tan 30°: 1√3 = √33. Calculator answers will usually give you the rationalised form — match it!
Exact values for any multiple of 30° or 45°
The unit circle’s symmetries let you find exact values for angles like 120°, 210°, 315°, etc. — and even angles bigger than 360° or negative angles. The trick: rewrite the angle as 180 ± θ or 360 ± θ where θ is a known angle.
🔍 The 4-step method
Rewrite your angle as 180 − θ, 180 + θ, 360 − θ, or 360 + θ where θ is an acute angle from the table.
Identify which quadrant your original angle lives in.
Use CAST to determine the sign (positive or negative).
Apply the symmetry rule, then read the magnitude off the table.
Example: sin 315°
315° = 360° − 45° → sin 315° = −sin 45° = −√22
Example: cos 210°
210° = 180° + 30° → cos 210° = −cos 30° = −√32
Example: tan 420°
420° = 360° + 60° → tan 420° = tan 60° = √3
For angles bigger than 360°, just subtract 360° as many times as needed until you land in [0°, 360°]. For negative angles, sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ.
Worked examples
WE 1
Derive exact values from an equilateral triangle (SME-style)
Using an equilateral triangle of side length 2 units, derive the exact values for sin, cos, and tan of π6 and π3.
Step 1: Split the equilateral triangle in half
Drop a perpendicular from the apex. You now have a right-angled triangle with hypotenuse 2 and base 1.
Step 2: Find the third side using Pythagorasa2 = 22 − 12 = 4 − 1 = 3a = √3Step 3: Apply SOH CAH TOAsin π6 = 12 | cos π6 = √32 | tan π6 = 1√3sin π3 = √32 | cos π3 = 12 | tan π3 = √3All six exact values derived ✓always start by sketching the triangle — it’s faster than memorising
WE 2
Find sin 150° as an exact value
Find the exact value of sin 150°.
Step 1: Rewrite 150° using a known angle150° = 180° − 30°Step 2: Identify the quadrant
150° is in quadrant 2. CAST → only sin is positive there. So sin 150° is positive.
Step 3: Apply the symmetry sin(180 − θ) = sin θsin 150° = sin(180 − 30) = sin 30° = 12sin 150° = 12positive sign + size from table = exact answer
WE 3
Find cos 225° as an exact value
Find the exact value of cos 225°.
Step 1: Rewrite using a known angle225° = 180° + 45°Step 2: Identify the quadrant
225° is in quadrant 3. CAST → only tan positive. So cos 225° is negative.
Step 3: Apply cos(180 + θ) = −cos θcos 225° = −cos 45° = −√22cos 225° = −√22
WE 4
Exact value in radians: tan(5π/6)
Find the exact value of tan 5π6.
Step 1: Convert / rewrite5π6 = π − π6 (equivalent to 180° − 30° = 150°)
Step 2: Identify the quadrant
150° → quadrant 2. CAST → only sin positive. So tan is negative.
Step 3: Apply tan(π − θ) = −tan θtan 5π6 = −tan π6 = −1√3tan 5π6 = −1√3 (or −√33 rationalised)
WE 5
Solve a trig equation using exact values
Solve cos θ = −12 for 0° ≤ θ ≤ 360°.
Step 1: Find the reference angle
From the table, cos 60° = 12. So the reference angle is 60°.
Step 2: Identify quadrants where cos is negative
Cos is negative in Q2 and Q3 (CAST: not A, not C).
Step 3: Find both solutions
Q2: 180 − 60 = 120°
Q3: 180 + 60 = 240°θ = 120° or 240°exact values + CAST = no calculator needed!
💡 Top tips
Sketch the two special triangles on your paper at the start of every Paper 1 — they take 5 seconds to draw and you can use them all exam.
Memorise the sine pattern (0, ½, √2/2, √3/2, 1). Cosine is the reverse, and tan = sin/cos.
Always rationalise the denominator in your final answer — examiners expect √2/2, not 1/√2.
For angles bigger than 90°, use the quadrant + reference angle approach: identify quadrant (sign), find reference angle (size), combine.
For radian questions, π = 180°. Convert if it helps you spot the angle, then convert back.
tan 90° is undefined, not 0 or ∞. If you get tan 90° in a question, something has gone wrong.
For angles like 480° or −60°, add or subtract 360° until you’re in [0°, 360°], then use exact values from there.
⚠ Common mistakes
Mixing up sin and cos at 30° and 60°. Remember: sin 30° = ½ (small angle, small sin); cos 30° = √3/2 (small angle, big cos).
Forgetting to rationalise. 1/√2 isn’t wrong, but examiners want √2/2 — and matching the markscheme is safer.
Wrong sign from CAST. Always identify the quadrant first, then the sign, then the size.
Treating tan 90° as 0 or 1 instead of “undefined”. Cos 90° = 0, so tan = sin/cos has no value there.
Using a calculator on Paper 1. You can’t! That’s why exact values are tested on the no-GDC paper.
Confusing reference angle with the original angle. The reference angle is always acute (0°–90°). Use it for size; use the quadrant for sign.
Mixing degree and radian forms in the same answer. If the question’s in radians, give the answer in radians.
Exact values are pure muscle memory — once you’ve drawn the two special triangles a dozen times, they become automatic. Combined with the unit circle and CAST, you can find the exact value of sin, cos, or tan for any multiple of 30° or 45° in seconds, without a calculator.
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