DSE Maths | Sine and Cosine Rule Lesson-Prep Guide (Tutor Edition) | Choosing the Formula and Solving Triangles

2026-11-01

Tutor lesson-prep guides · Secondary Maths

Students can recite all three formulas and still substitute into the wrong one. This lesson turns formula choice into a two-step check: is there an opposite pair, or an included angle?

Objectives — Students can: (1) use the cosine rule to find a side from two sides and the included angle, or an angle from three sides; (2) use the sine rule to find sides from one side and two angles and handle the ambiguous case; (3) find area with half ab sin C and with Heron formula.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Trig ratios	What is sin 60 degrees	Root three over two, about 0.866
Inverse sine	sin B = 0.625: how many angles	Two, one acute and one obtuse
Angle sum	What do the three angles add to	180 degrees

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Choosing the formula	Two questions side by side, look for an opposite pair	Name the formula
8-22 min	Cosine rule	c squared = a squared + b squared – 2ab cos C with 7, 8, 60 degrees	c = 7.55
22-34 min	Sine rule	a over sin A = b over sin B, solve for c	c = 14.10
34-44 min	Area	half ab sin C and Heron with 13, 14, 15	24.25 and 84
44-54 min	Ambiguous case	a = 8, b = 10, A = 30 degrees	B = 38.7 or 141.3
54-60 min	Exit check	Three questions collected and marked	Done independently

03 Teaching script (read it out)

Scene 1: Decide first, substitute second

Two checks: (1) is there a matching side and opposite angle? Then the sine rule applies. (2) are there two sides with the included angle, or three sides? Then the cosine rule applies. The cosine rule is Pythagoras plus a correction term: at 90 degrees cos C is zero and it collapses to a squared plus b squared.

Task: for three textbook questions, write only which formula applies and why, no arithmetic.

Scene 2: The arithmetic of both rules

(1) Sides 7 and 8 with included angle 60 degrees: c squared = 113 minus 112 times 0.5, which is 57, so c = 7.550. (2) Sine rule: A = 180 – 40 – 65 = 75 degrees, and c = 10 sin 65 over sin 40 = 14.10.

Task: explain why both routes must give the same side length, and what a mismatch usually means.

Scene 3: Area and the ambiguous case

Area has two routes: half ab sin C = 24.25, or Heron when only three sides are known, with s = 21 giving 84. In the ambiguous case with a = 8, b = 10 and A = 30 degrees, sin B = 0.625, so B is 38.7 or 141.3 degrees and both must fit the angle sum.

Warn: a question asking for B expects both solutions plus a check that the angles still sum below 180.

04 Question types and worked steps

Type	Example	Steps	Answer
Calculation	Side from two sides and included angle	c squared = a squared + b squared – 2ab cos C	c = 7.55
Calculation	Angle from three sides	cos C = (a squared + b squared – c squared) over 2ab	Invert cosine last
Calculation	Side from one side and two angles	a over sin A = b over sin B	c = 14.10
Calculation	Area	half ab sin C or Heron	24.25 or 84

Sanity check (teach this): The longest side faces the largest angle, and any side must be shorter than the sum of the other two (7.55 is below 15). Area can never exceed half ab, which is 28.

05 Independent practice (3 questions)

Question	Answer	What to watch
sin B for a = 8, b = 10, A = 30 degrees	0.625	Pair b with B and a with A
Area of the triangle with sides 13, 14, 15	84	Heron, semi-perimeter first
Circumradius of the same triangle	8.125	R = abc over four times area

06 Common mistakes and what to say

Mistake	Student writes	Say this
Sign placed on the wrong square	Writing minus a squared in the cosine rule	“The side opposite the angle is the one subtracted.”
Angle entered in radians	cos 60 read as 0.999	“Check that DEG appears on the calculator.”
Only one ambiguous solution	Answering just B = 38.7	“sin B = 0.625 gives two angles; write both.”
Forgetting the half	Answering 48.50 for area	“Area is half ab sin C.”

07 Homework (paper, section and question numbers)

Secondary maths chapter index — All chapters
DSE maths practice paper — Trigonometry and solving triangles

08 Exit check (last 5 minutes)

Question	Answer	Decision
c for a = 7, b = 8, C = 60 degrees	7.55	All three right: move to statistics
Area of the 13/14/15 triangle	84	One wrong: redo Heron before next lesson
Possible values of B in the ambiguous case	38.7 or 141.3	Two wrong: restart act three

09 Tutor FAQ

Must both rules be memorised?

Yes, but the habit that matters is the first check for an opposite pair; the formula then follows.

When is Heron mandatory?

Only when three sides are given with no angle, otherwise half ab sin C is faster.

Does the ambiguous case appear every year?

Occasionally, and when it does it carries the discriminating marks.

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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.