DSE Maths | Trigonometric Ratios and Bearings Lesson-Prep Guide (for tutors) | sin cos tan, cosine rule, three-figure bearings

2026-11-03

Tutor lesson-prep guides · Secondary Maths

Students can recite SOH CAH TOA but freeze at “the bearing of B from A”. This lesson turns bearings into an angle difference, then decides between right-triangle work and the cosine rule.

Objectives — Students can: (1) choose the correct ratio from the given sides and angle; (2) turn “from A” and “from B” bearings into two three-figure bearings and find the included angle; (3) use the cosine rule when there is no right angle, and say when sin/cos/tan cannot be used.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Pythagoras	Legs 3 and 4, hypotenuse?	5
Angle mode	Which mode should the calculator be in?	Degree (DEG)
Bearing definition	Measured clockwise from north, what range?	000 to 360 degrees

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Choose a ratio	Match known and wanted sides	Three oral choices
8-20 min	Slope example	Angle 25 degrees, slope length 120 m	Rise = 50.7 m
20-34 min	Bearings	From A versus from B, sketch the turn	Included angle 100 degrees
34-46 min	Cosine rule	Third side with a non-right angle	Distance 11.65 km
46-54 min	Answer format	Sketch, formula, substitution, units, 3 s.f.	Rewrite one answer
54-60 min	Exit check	Three questions collected and marked	Independent

03 Teaching script (read it out)

Scene 1: Label first, then choose

Mark the angle wanted, then label each side: opposite, adjacent, hypotenuse. Two sides and an angle wanted means an inverse ratio; one side and one angle means multiply or divide directly.
A slope of 25 degrees with a slope length of 120 m: sin 25 = rise / 120, so rise = 120 x 0.423 = 50.7 m.

Ask the student: which ratio gives the horizontal run here (cosine, 108.8 m)?

Scene 2: Two bearings are one side plus one angle

Bearings are always measured clockwise from north and written with three digits.
“Bearing of B from A is 040” means walk 6 km from A on 040; “bearing of C from B is 140” means walk 9 km from B on 140.
The angle between the two legs is 140 – 40 = 100 degrees, because both were measured from the same reference direction.

Ask the student: state the bearing of B from C (220 degrees, add 180 then take the remainder).

Scene 3: The cosine rule only when the angle is not 90 degrees

a squared = b squared + c squared – 2bc cos A. With b = 6, c = 9 and A = 100 degrees: a squared = 36 + 81 – 108 x cos 100 = 135.75, so a = 11.65 km.
At 90 degrees cos is zero and the rule collapses into Pythagoras, so there is nothing extra to memorise.

Remind: “shortest distance” means the straight line between the two points, not the sum of the legs.

04 Question types and worked steps

Type	Example	Steps	Answer
Find a side	One angle and the hypotenuse	Write the ratio, rearrange, compute	50.7 m
Find an angle	Two sides given	Label, choose, inverse function	about 36.9 degrees
Bearings	Two legs of a route	Sketch, subtract for the included angle	100 degrees
Non-right triangle	Two sides and included angle	Cosine rule, then square root	11.65 km

Sanity check (teach this): The hypotenuse is always the longest side, so a computed opposite can never exceed it, and sin or cos values always stay below 1. An included angle between bearings must lie between 0 and 180 degrees; a result like 260 means the wrong direction was taken, so subtract from 360.

05 Independent practice (3 questions)

Question	Answer	What to watch
Rise for a 25-degree slope of 120 m	50.7 m	Did they use sine, not cosine?
Distance between the two bearing legs	11.65 km	Is the included angle 100 degrees?
Bearing of B from C	220 degrees	Add 180 then take the remainder

06 Common mistakes and what to say

Mistake	Student writes	Say this
Swapping opposite and adjacent	sin and cos reversed, answer too large	“Which side is actually touching the angle?”
Calculator left in RAD	sin 25 returns 0.42	“Check DEG before any angle calculation.”
Adding 180 to both bearings	Included angle becomes 280	“Both are from north, so just subtract.”
Using tan without a right angle	tan applied to the whole triangle	“No right angle means no opposite or adjacent: use the cosine rule.”

07 Homework (paper, section and question numbers)

DSE Maths practice paper (trigonometry and measurement) — Section B question 14: bearings and cosine rule
Secondary maths chapter index — All chapters

08 Exit check (last 5 minutes)

Question	Answer	Decision
Rise for a 25-degree slope of 120 m	50.7 m	Three correct means move on to the sine rule
Straight-line distance from the two bearings	11.65 km	One wrong means reteach the included angle
When tan cannot be used	When there is no right angle	Two wrong means restart act one

09 Tutor FAQ

Must students memorise ratio values?

No, the calculator gives them; but sin 30 = 0.5 and cos 60 = 0.5 are worth knowing for a quick check.

Can bearings be written as 60 degrees?

The Hong Kong syllabus uses three digits, so 060 is clearer and always accepted.

Cosine rule or sine rule?

Two sides with the included angle, or three sides, means cosine; two angles with a side means sine.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.