IP EMaths Notes (Upper Sec, Year 3-4): 10) Trigonometric Applications

Study guide08 Nov 2025, 00:00 ZUpdated 17 Jul 2026

Apply sine and cosine rules, bearings, and area formulas to non-right-angled triangles and navigation tasks.

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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.
Exam-track route: use the separate O-Level and SEC G3 Mathematics notes for K310 topic ownership.

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 $\tfrac{1}{2} ab \sin C$

Reviewed by

Marcus Pang·Managing Director (Maths)

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.

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	$\Delta = \tfrac{1}{2} ab \sin C$	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

What the diagram gives	First move	Why it works	Common trap
Two sides with their included angle	Substitute directly into $\tfrac{1}{2}ab\sin C$ .	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 $\sin C$ 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.

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.

IP EMaths Notes (Upper Sec, Year 3-4): 10) Trigonometric Applications

How this chapter applies

Sources

Formula summary

Obtuse-angle checkpoint

Formula Selection Map

Area-formula setup checkpoint

Worked example - Cosine rule for an unknown side

Worked example - Sine rule to find an angle

Ambiguous-case trap

Bearing-angle checkpoint

Worked example - Bearings using the cosine rule

Practice Quiz

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