S4 Maths: Sine and Cosine Rule — Tutor Lesson Plan (45 min, side and angle choice, four errors)

2026-11-02

Tutor lesson-prep guides · Secondary mathematics (DSE)

The rules are easy to state and hard to choose. Students apply the cosine rule to a triangle with no included angle, or use the sine rule to find the largest side. This lesson builds the choice first.

Objectives — the student can pick the sine rule, the cosine rule or the area formula from the given information alone, apply the cosine rule to find a side or an angle, handle the ambiguous case of the sine rule, and check an answer against the angle-side ordering of the triangle.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Right-triangle trigonometry	What is sin 30?	0.5
Angle sum	Two angles are 40 and 60. What is the third?	80 degrees
Rearranging	Make b the subject of a over sin A equals b over sin B	b = a sin B / sin A

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Hook	Give a triangle with two sides and the included angle	Decide what can be found
5-13 min	Scene 1	Area equals one half ab sin C	Get 12 from 6, 8 and 30 degrees
13-24 min	Scene 2	Cosine rule for the included angle case	Get a = 4.11
24-33 min	Scene 3	Sine rule for the opposite-pair case	Get b = 10.8
33-40 min	Scene 4	The ambiguous case and when to ignore it	Name the second possible angle
40-45 min	Exit check	Three items in section 08	Marked at once

03 Teaching script (read it out)

Scene 1: Choose before you compute

Two sides and the included angle: cosine rule. Three sides: cosine rule, rearranged for the angle. A side with its opposite angle: sine rule. Two sides and a non-included angle: sine rule, and expect the ambiguous case. Write these four lines on the board and make students say which line they are on before touching the calculator.

Ask: “Sides 6 and 8, angle 30 between them. Which rule?” The answer is cosine rule, and the students who say sine rule are matching words, not structure.

Scene 2: The cosine rule, both directions

a squared equals b squared plus c squared minus 2bc cos A. With b = 6, c = 8 and A = 30 degrees: 36 plus 64 minus 96 times 0.866, which is 100 minus 83.1, so a squared is 16.9 and a is 4.11. Rearranged for the angle: cos A equals b squared plus c squared minus a squared, all over 2bc. Learn it as one formula plus a rearrangement, not two formulas.

Warn: the 2bc term is subtracted. Students adding it get 183 and a side of 13.5, which is longer than both given sides and should be caught by the sanity check.

Scene 3: The sine rule and the ambiguous case

a over sin A equals b over sin B. With a = 8, A = 40 and B = 60: b equals 8 sin 60 over sin 40, which is 10.8. When the given angle is not between the two given sides, the sine of the supplement gives a second valid triangle. Both answers exist unless the diagram or the angle sum rules one out: compute the third angle for each and discard the one that makes the sum exceed 180.

Ask: “Sine gives 50 degrees. What is the other angle with the same sine?” The answer is 130, and the angle sum is what tells you whether it survives.

04 Question types and worked steps

Type	Example	Steps	Answer
Area	Find the area of a triangle with sides 6 and 8 and included angle 30.	One half ab sin C	12
Side	Find a when b = 6, c = 8 and A = 30.	Cosine rule, then square root	4.11
Angle	Find A when a = 7, b = 6, c = 8.	Rearranged cosine rule	about 60.4 degrees
Side	Find b when a = 8, A = 40 and B = 60.	Sine rule	10.8

Sanity check (teach this): Every answer must survive three checks: the largest angle faces the largest side, the three angles add to 180, and each side is shorter than the sum of the other two. A computed side longer than both given sides in an included-angle question means the cosine rule was applied with the wrong sign. For an angle, a cosine outside minus 1 to 1 means the side lengths were substituted in the wrong order.

05 Independent practice (3 questions)

Question	Answer	What to watch
Find the area of a triangle with sides 7 and 9 and included angle 45.	One half times 7 times 9 times sin 45 = 22.3.	Degrees, not radians
Find the third side when b = 7, c = 9 and A = 45.	a squared = 49 plus 81 minus 126 cos 45 = 40.9, so a = 6.40.	Subtract the 2bc term
Find the angle opposite the side of length 10 in a triangle with sides 6, 8 and 10.	cos A = (36 plus 64 minus 100) over 96 = 0, so A = 90 degrees.	Sanity check: 6-8-10 is right angled
A triangle has a = 5, b = 7 and A = 30. How many triangles fit?	Two: the ambiguous case gives a second angle near 44.4 or 135.6 degrees, and the second survives if the sum stays under 180.	Check the angle sum for both

06 Common mistakes and what to say

Mistake	Student writes	Say this
Applying the cosine rule without an included angle	The cosine rule needs the angle between the two known sides	Ask the student to point at the angle and name the two sides it touches
Adding the 2bc cos A term	It is subtracted; only the included angle case can give a side shorter than both givens	Ask for the sanity check before the final line
Mixing degrees and radians	The calculator mode is the first thing to set, not the last	Ask for the mode on screen before any trig value is read
Forgetting the ambiguous case	A sine answer of 0.766 gives both 50 and 130 degrees, and the angle sum decides	Ask how many triangles the data allow before solving

07 Homework (paper, section and question numbers)

HKDSE Mathematics Compulsory Part Paper 1 practice paper — Triangle and trigonometry questions; redo section 05 questions 1 to 4
Secondary mathematics chapter index — Read this chapter with the trigonometry chapter

08 Exit check (last 5 minutes)

Question	Answer	Decision
Sides 5 and 7, angle 60 between them. Which rule finds the third side?	Cosine rule	All correct: move on
A sine rule answer gives sin B = 0.643. Give both possible angles.	40 and 140 degrees	One answer means the ambiguous case is still missing
Find the area with sides 4 and 5 and included angle 90.	10 square units	Half of 4 times 5

09 Tutor FAQ

Is the sine rule ever needed when the triangle is right angled?

It still works, but the basic ratios are faster. Use the sine rule when the triangle has no right angle to lean on.

Must students memorise the rearranged cosine rule?

They should be able to derive it in two lines from the standard form. Memorising both invites mixing them up.

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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.