For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.
How this chapter applies
Eclat core: obtuse-angle sine and cosine, triangle area, sine rule, cosine rule, bearings, elevation, depression, and ambiguous-case reasoning form the main route.
School-sensitive extension: proof-heavy modelling and more demanding navigation chains may vary by school.
2027 national comparison: K310 Topic G4 explicitly extends sine and cosine to obtuse angles and includes the triangle-area formula, sine rule, cosine rule, bearings, and two-dimensional or three-dimensional applications.
Check your school: confirm bearing notation, ambiguous-case expectations, and rounding conventions.
Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 10) Trigonometric Applications cover? A: Apply sine and cosine rules, bearings, and area formulas to non-right-angled triangles and navigation tasks.
The core idea is simple: When a triangle is not right-angled, use sine rule, cosine rule, or area formula.
Use it as a working check: Sketch first. Use cosine rule for two sides with the included angle, sine rule for matched side-angle pairs, and bearing differences to find the angle between paths.
Then go one layer deeper: Follow the side, angle, and bearing examples to practise a clean decision process: draw the triangle, name the known values, choose the formula, then check whether any second angle is possible.
Extend beyond right triangles with the sine rule, cosine rule, and 21absinC
Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.
K310 extends sine and cosine to obtuse angles. Between 90 degrees and 180 degrees, sine is positive and cosine is negative. This sign check matters when a calculator answer or a cosine-rule rearrangement produces an angle.
Situation
Result
Check
Supplementary angles in a sine calculation
They have the same sine value.
Test whether both angles fit the triangle and the stated data.
Cosine of an obtuse angle
The value is negative.
Keep the negative value until applying inverse cosine.
Cosine rule gives a negative cosine
The included angle is obtuse.
Check that the longest side is opposite the largest angle.
Misconception check: the ambiguous case belongs to the sine rule. Do not invent a second triangle when its side lengths or angle sum make it impossible.
Sine rule: sinAa=sinBb=sinCc.
Cosine rule: c2=a2+b2−2abcosC.
Area: Δ=21absinC.
Formula Selection Map
Use this map before substituting numbers. Most errors happen because the correct triangle was drawn, but the formula was chosen from habit instead of from the known side-angle pattern.
What you know from the sketch
Best first move
Why it works
Two sides and the included angle
Cosine rule for the third side
The included angle sits between the two known sides
Three sides
Cosine rule rearranged for an angle
No angle-side pair is available for the sine rule
One matching side-angle pair and one more side/angle
Sine rule
A complete opposite pair anchors the proportion
Two sides with a non-included angle
Sine rule, then check the ambiguous case
A second possible angle may fit the same sine value
Two sides and the included angle, area needed
Δ=21absinC
The formula needs the angle between the two selected sides
Bearing paths from the same point
Difference of bearings, then cosine rule
The bearing difference gives the included angle between paths
Misconception check: do not use the sine rule just because one angle is visible. You need a complete opposite side-angle pair, such as a with A, before the proportion is usable.
Area-formula setup checkpoint
Use Δ=21absinC only when the angle is between the two selected sides. If the given angle is not included, find the included angle first or choose a different method.
What the diagram gives
First move
Why it works
Common trap
Two sides with their included angle
Substitute directly into 21absinC.
The formula measures the height created by the angle between those sides.
Using the angle opposite one of the sides instead.
Two sides and a non-included angle
Find the missing included angle first, often with sine rule or angle sum.
The area formula needs the angle between the two sides being multiplied.
Multiplying the two sides and any available angle without checking its position.
Three sides only
Use cosine rule to find one angle, then use the area formula.
Once an included angle is known, the area formula becomes usable.
Trying to use sinC before any angle has been found.
Bearing triangle with two distances from the same point
Use the bearing difference as the included angle.
Both distances start from the same point, so the angle between tracks is included.
Using one of the original bearings as the triangle angle.
Worked check: if AB=8cm, AC=5cm, and ∠BAC=40∘, then the angle is included between AB and AC. The area is
Δ=21(8)(5)sin40∘≈12.9cm2.
Misconception check: the area formula is not "any two sides and any angle". It is "two sides and the angle between them".
Worked example - Cosine rule for an unknown side
In triangle ABC, sides AB=7.5cm and AC=6.2cm with included angle ∠BAC=58∘. Find the length of BC.
Apply the cosine rule, working in centimetres:
BC2=(7.5)2+(6.2)2−2(7.5)(6.2)cos(58∘).
Square the sides: 7.52=56.25 and 6.22=38.44 (both in cm2).
Evaluate the product term:
2(7.5)(6.2)cos(58∘)≈93.0×0.5299≈49.28.
Substitute the values back into the equation:
BC2≈56.25+38.44−49.28=45.41.
Take the square root to find the side length:
BC=45.41≈6.74.
Therefore BC≈6.74cm.
Worked example - Sine rule to find an angle
In triangle XYZ, side x=9.5cm is opposite angle X=53∘, and side y=7.2cm is opposite angle Y. Find Y to the nearest degree.
Apply the sine rule: sinXx=sinYy.
Substitute the known values: Result:sin53∘9.5=sinY7.2
Rearrange for sinY: Result:sinY=9.57.2sin53∘≈9.57.2×0.7986≈0.605
Take the inverse sine: Y≈sin−1(0.605)≈37.3∘.
Check for the supplementary angle: 180∘−37.3∘=142.7∘, but 53∘+142.7∘>180∘
Angle Y is approximately 37∘.
Ambiguous-case trap
The sine rule can produce two angles because sinθ=sin(180∘−θ). Always test the supplementary angle before finalising an answer:
Find the calculator angle from inverse sine, sin−1.
Find the possible supplement, 180∘−θ.
Add it to the other known angle in the triangle.
Reject it if the angle sum is 180∘ or more.
In the example above, 142.7∘ is rejected because 53∘+142.7∘>180∘. That is why the smaller angle is the only valid one.
Bearing-angle checkpoint
For bearing questions, decide which angle is inside the triangle before choosing a trigonometric rule. Bearings are measured clockwise from north, but the triangle angle may be a bearing difference, a turn angle, or an angle found after reversing a direction.
Diagram clue
First angle move
Why this is the triangle angle
Common trap
Two paths start from the same point
Subtract the smaller bearing from the larger bearing.
Both directions are measured from the same north line at the same point.
Treating the two bearings as separate triangle angles.
A route goes from A to B, then from B to C
Reverse the bearing of BA if the given bearing is AB.
The angle at B uses the direction from B back to A and the direction from B to C.
Using the outward bearing from A at point B.
A bearing differs by more than 180 degrees
Use the smaller angle between the two directions if the triangle needs an interior angle.
Interior triangle angles must be less than 180 degrees.
Putting the reflex angle into the cosine rule.
Parallel north lines are drawn at different points
Copy equal corresponding angles between the parallel north lines.
North lines on a bearing diagram are parallel, so alternate and corresponding angles can transfer.
Assuming only one north arrow can be used.
Misconception check: a bearing is a direction from a point, not automatically an interior angle of the triangle. Mark the point where the triangle angle sits before subtracting or reversing bearings.
Worked example - Bearings using the cosine rule
Two ships leave a harbour at the same time. Ship P sails 45km on a bearing of 040∘ while ship Q sails 52km on a bearing of 120∘. Estimate their distance apart after the journey.
Sketch bearings from the harbour (taken as the origin). The angle between the tracks is the difference in bearings: 120∘−40∘=80∘.
Model the situation as a triangle with sides 45 km and 52 km, with included angle 80∘.
Apply the cosine rule for the separation d: Result:d2=452+522−2(45)(52)cos80∘
Compute:
452=2025,
522=2704,
2(45)(52)=4680
Substitute: d2≈2025+2704−812.7=3916.3.
Take the square root: d≈3916.3≈62.6km.
The ships are approximately 62.6km apart after their journeys.
Practice Quiz
Check sine- and cosine-rule fluency, area calculations, and navigation bearings before moving on.
