🔵 Circles & Trigonometry
This chapter covers circle properties, angle properties of circles, Pythagoras' theorem, trigonometric ratios (SOH CAH TOA), sine rule, cosine rule, area of triangle, and bearings. These topics carry very heavy marks in Paper 2 and require both conceptual understanding and computational accuracy.
Circle theorems are a favourite topic for examiners because they test geometric reasoning. Each theorem must be quoted precisely in your working. Trigonometry, on the other hand, is a computational skill — you need to know when to use SOH CAH TOA (right-angled triangles only) versus the sine rule or cosine rule (any triangle).
Circle Theorems
Circle theorems describe the relationships between angles, chords, and tangents in a circle. There are six main theorems you must know. In the exam, you must state the theorem name and apply it correctly. Many questions combine two or three theorems in a single problem.
The angle at the centre is twice the angle at the circumference subtended by the same arc
6 Essential Circle Theorems
Angle at centre = 2 x angle at circumference
The angle subtended by an arc at the centre of the circle is exactly twice the angle subtended at any point on the circumference, from the same arc.
Angle in a semicircle = 90 degrees
Any angle inscribed in a semicircle (i.e., subtended by a diameter at the circumference) is always a right angle.
Angles in same segment are equal
All angles subtended by the same arc (or chord) at points on the same side of the chord are equal.
Opposite angles of cyclic quadrilateral sum to 180 degrees
A cyclic quadrilateral has all four vertices on the circle. Its opposite angles are supplementary.
Tangent is perpendicular to radius
A tangent to a circle is perpendicular (90 degrees) to the radius drawn to the point of tangency.
Tangents from external point are equal
Two tangent lines drawn from the same external point to a circle have equal length.
⭕Circle Theorems
Essential circle properties tested in O-Level E-Math. Must know all theorems and their converses.
📝Worked Example 1: Angle at Centre Theorem
In the circle with centre O, angle AOB = 124 degrees. C is a point on the major arc. Find angle ACB.
Angle at centre is twice the angle at the circumference
📝Worked Example 2: Cyclic Quadrilateral
ABCD is a cyclic quadrilateral. Angle A = 110 degrees and angle B = 85 degrees. Find angles C and D.
Opposite angles of a cyclic quadrilateral sum to 180°
📝Worked Example 3: Tangent-Radius Problem
A tangent touches a circle with centre O at point T. OT = 5 cm and the tangent meets a line from O at point P, where OP = 13 cm. Find PT.
Tangent is perpendicular to radius at the point of contact
📝Worked Example 3b: Angles in the Same Segment
Points A, B, C, and D lie on a circle. AC and BD intersect at E inside the circle. Angle AEB = 70 degrees and angle ABD = 40 degrees. Find angle ACD.
The angle at the centre is twice the angle at the circumference subtended by the same arc
📝Worked Example 3c: Alternate Segment Theorem
A tangent at point T makes angle 55 degrees with chord TA. B is a point on the circle on the opposite side of chord TA from the tangent. Find angle ABT.
Alternate segment theorem: both angles equal 55°
📝Worked Example 3d: Multi-Theorem Circle Problem
ABCD is a cyclic quadrilateral. The tangent at A makes angle 50 degrees with AB. Angle ADC = 115 degrees. Find (a) angle ACB, (b) angle ABC.
Opposite angles of a cyclic quadrilateral sum to 180°
Circle Theorem Exam Language
Examiners award marks for precise theorem names. Use these exact phrasings:
- "Angle at centre = 2 x angle at circumference"
- "Angle in a semicircle = 90 degrees"
- "Angles in the same segment are equal"
- "Opposite angles of a cyclic quadrilateral are supplementary"
- "Tangent perpendicular to radius"
- "Tangents from an external point are equal in length"
- "Alternate segment theorem"
Do NOT use vague language like "the angles are the same" or "they add up to 180". Be specific about WHICH theorem you are using.
A tangent at point P makes angle 65 degrees with chord PQ. R is on the circle in the alternate segment. What is angle PRQ?
Pythagoras' Theorem & Trigonometric Ratios
Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This is the foundation for all distance calculations in coordinate geometry and many real-world problems. Trigonometric ratios (SOH CAH TOA) relate the angles and sides of right-angled triangles.
📏Pythagoras Theorem
Relates the sides of a right-angled triangle. Foundation for trigonometry and coordinate geometry.
📊Trigonometric Ratios (SOH CAH TOA)
For right-angled triangles. The three primary trigonometric ratios.
📝Worked Example 4: SOH CAH TOA
In right triangle PQR, angle Q = 90 degrees, PQ = 8 cm, and angle P = 35 degrees. Find QR and PR.
Right triangle PQR: identify opposite, adjacent and hypotenuse relative to angle P
📝Worked Example 5: Pythagoras in 3D
A rectangular box has length 12 cm, width 4 cm, and height 3 cm. Find the length of the space diagonal (from one corner to the opposite corner through the interior).
Cuboid with base diagonal AC and space diagonal AG
📝Worked Example 5b: Angle of Elevation
A boy stands 50 m from the base of a building. The angle of elevation to the top is 32 degrees. Find the height of the building.
Angle of elevation from a point on the ground to the top of a building
📝Worked Example 5c: Angle of Depression
From the top of a cliff 80 m high, the angle of depression to a boat at sea is 25 degrees. How far is the boat from the base of the cliff?
Angle of elevation from a point on the ground to the top of a building
📝Worked Example 5d: Finding an Angle Using Inverse Trig
A ladder 5 m long leans against a wall. The foot of the ladder is 3 m from the wall. Find (a) how high up the wall the ladder reaches, and (b) the angle the ladder makes with the ground.
Right triangle PQR: identify opposite, adjacent and hypotenuse relative to angle P
Exact Trigonometric Values
You must memorise the exact values for the standard angles: 0 degrees, 30 degrees, 45 degrees, 60 degrees, and 90 degrees. These come up frequently in both Paper 1 (non-calculator) and Paper 2. The examiners expect exact values (surds, not decimals) when these special angles appear.
| Angle | sin | cos | tan |
|---|---|---|---|
| undefined |
Memory Tricks for Exact Values
For sine of 0°, 30°, 45°, 60°, 90°: the numerators follow the pattern , all divided by 2. This gives . For cosine, it is the same sequence but reversed. For tan, simply divide sin by cos for each angle.
SOH CAH TOA — How to Remember
Some Old Horse Caught Another Horse Taking Oats Away
Or simply: SOH-CAH-TOA (say it like a word: "so-ka-toa")
- SOH:
- CAH:
- TOA:
Exam trick: To decide which ratio to use, identify which two of the three sides (Opposite, Adjacent, Hypotenuse) are involved in your question. The ratio that uses those two sides is the one you need.
What is as an exact value?
Sine (blue) and cosine (red) curves with period 360° and amplitude 1
Sine Rule & Cosine Rule
SOH CAH TOA only works for right-angled triangles. For any other triangle, you need the sine rule or cosine rule. The choice depends on what information you have.
When to Use Sine Rule vs Cosine Rule
Use Sine Rule when you have a matched pair (a side and its opposite angle) plus one more piece. Use Cosine Rule when you have SAS (two sides and the included angle) or SSS (all three sides). If the triangle has a right angle, just use SOH CAH TOA — it is simpler.
🔺Sine Rule & Cosine Rule (Non-Right Triangles)
For any triangle (not just right-angled). Used when SOH CAH TOA cannot be applied directly.
📝Worked Example 6: Cosine Rule (Finding a Side)
In triangle ABC, b = 8 cm, c = 5 cm, angle A = 60 degrees. Find side a.
📝Worked Example 7: Cosine Rule (Finding an Angle)
In triangle PQR, p = 7, q = 8, r = 13. Find angle R.
📝Worked Example 8: Sine Rule (Finding an Angle)
In triangle PQR, p = 10, q = 7, angle P = 50 degrees. Find angle Q.
📝Worked Example 9: Area Using Sine
Find the area of triangle ABC where a = 12, b = 9, and angle C = 40 degrees.
📝Worked Example 9b: Sine Rule (Finding a Side)
In triangle ABC, angle A = 42 degrees, angle B = 73 degrees, and side a = 15 cm. Find side b.
📝Worked Example 9c: Real-World Cosine Rule Application
Two ships leave port P at the same time. Ship A travels on bearing 060 degrees at 15 km/h for 2 hours. Ship B travels on bearing 150 degrees at 10 km/h for 2 hours. Find the distance between the ships after 2 hours.
Bearing: P to A (060°), A to B (150°)
The Ambiguous Case of Sine Rule
When using the sine rule to find an angle, there may be two possible answers: one acute and one obtuse. This happens because sin x = sin(180 - x). You must check whether the obtuse angle is valid by adding it to the known angles and checking if the sum is less than 180 degrees. If both are valid, state both solutions. In O-Level exams, the question usually specifies "acute" or "obtuse", or the diagram makes it clear which one to use.
Decision Flowchart: Which Formula to Use?
Is the triangle right-angled?
YES: Use SOH CAH TOA (for sides/angles) or Pythagoras (for sides only). These are simpler and should be preferred.
Do you have a matched pair (side + opposite angle)?
YES: Use Sine Rule. You need at least one matched pair plus one extra piece of information.
Do you have SAS (two sides + included angle)?
YES: Use Cosine Rule to find the third side. Or use the area formula (1/2 ab sin C) for the area.
Do you have SSS (all three sides)?
YES: Use Cosine Rule to find any angle.
Do you need the area?
Use Area = 1/2 ab sin C if you have SAS. Otherwise, use Area = 1/2 base x height if you know the base and height.
In triangle ABC, a = 10, angle A = 30 degrees, angle B = 45 degrees. Which formula should you use to find side b?
Bearings
Bearings are measured clockwise from North and always written as 3-digit numbers (e.g., 045 degrees, 130 degrees, 270 degrees). The bearing of B from A means: stand at A, face North, then measure the angle clockwise to the direction of B. To find the back bearing (bearing of A from B), add or subtract 180 degrees.
🧭Bearings & Angles of Elevation/Depression
Bearings are measured clockwise from North. Angles of elevation/depression are measured from the horizontal.
📝Worked Example 10: Bearing Problem
A ship sails from P on bearing 040 degrees for 80 km to Q. It then sails on bearing 130 degrees for 60 km to R. Find PR and the bearing of R from P.
Bearing: P to Q (040°), Q to R (130°)
📝Worked Example 10b: Three-Point Bearing with Cosine Rule
From point A, point B is on bearing 060 degrees at distance 8 km. From A, point C is on bearing 150 degrees at distance 10 km. Find (a) angle BAC and (b) distance BC.
Bearing: A to B (060°), B to C (150°)
📝Worked Example 10c: Non-Right Bearing Triangle
A hiker walks from P to Q on bearing 075 degrees for 6 km, then from Q to R on bearing 200 degrees for 4 km. Find the distance PR and the bearing of R from P.
Bearing: P to Q (075°), Q to R (200°)
Bearing Problem Checklist
Follow this checklist for every bearing question:
- Draw a clear diagram with North lines at EVERY point
- Mark all given bearings and distances
- Find angles inside the triangle using back bearings (add or subtract 180 degrees)
- Use the appropriate formula: Pythagoras, sine rule, cosine rule
- Express the final bearing as a 3-digit number (e.g., 076 degrees, not 76 degrees)
The bearing of B from A is 310 degrees. What is the bearing of A from B?
⚠️Common Mistakes — Circles & Trig
Why: Sine rule: need a side-angle pair. Cosine rule: need SAS or SSS. Using the wrong rule gives incorrect answers.
Why: Bearings are ALWAYS written as 3-digit numbers, measured clockwise from North. Use leading zeros: 045, 090, 005.
Why: Area uses r-squared, circumference uses 2r. Remember: area = pi r^2, circumference = 2 pi r.
Why: A cone is one-third of a cylinder with the same base and height. Forgetting the 1/3 is a frequent error.
Why: The coefficient is 4, not 2. Hemisphere curved SA = 2 pi r^2, but a full sphere is 4 pi r^2.
3D Trigonometry Problems
Three-dimensional trigonometry questions involve finding lengths or angles in solid shapes such as cuboids, pyramids, and prisms. The technique is always the same: identify right-angled triangles within the 3D shape and apply Pythagoras or SOH CAH TOA to each triangle separately. You usually need to find an intermediate length (such as a diagonal of a face) before you can find the required answer.
📝Worked Example 11: Angle in a Cuboid
A cuboid has dimensions 6 cm by 8 cm by 5 cm. Find the angle that the space diagonal AG makes with the base ABCD.
Cuboid with base diagonal AC and space diagonal AG
Square-based pyramid with VO = 12 cm, OM = 5 cm, VM = 13 cm
📝Worked Example 12: Angle in a Pyramid
A square-based pyramid has base side 10 cm and vertical height 12 cm. Find (a) the slant height and (b) the angle between a triangular face and the base.
Strategy for 3D Trigonometry
Step 1: Draw a clear 3D diagram
Label all given measurements. Use dotted lines for hidden edges.
Step 2: Identify the right-angled triangles
Most 3D trig problems reduce to 2 or 3 right-angled triangles. The triangles are usually on the faces or cross-sections of the solid.
Step 3: Find intermediate lengths first
You often need to find a base diagonal or face diagonal before you can find the final angle or length.
Step 4: Apply Pythagoras or SOH CAH TOA
Work through each right triangle separately. State which triangle you are working with.
Step 5: State the required angle clearly
Describe the angle using three letters (e.g., angle GAC) or a clear verbal description.
A cuboid is 3 cm by 4 cm by 12 cm. Find the length of the space diagonal.
Cosine Rule: Finding an Angle
The cosine rule can be rearranged to find an angle when all three sides are known. This is a very common Paper 2 question. Remember to use the rearranged form: . The side opposite the angle you are finding goes in the numerator with a MINUS sign.
📝Worked Example 13: Cosine Rule for Angle
In triangle PQR, PQ = 7 cm, QR = 9 cm, PR = 11 cm. Find angle PQR.
📝Worked Example 14: Area of Triangle Using Sine
In triangle ABC, AB = 8 cm, AC = 12 cm, angle BAC = 53 degrees. Find the area of the triangle.
In triangle ABC, all three sides are known but no angles. To find angle B, which formula do you use?