S4 Maths | Trigonometry prep guide (for tutors) | sin cos tan, sine rule, cosine rule

2026-11-01

Tutor lesson-prep guides · Secondary Mathematics

Students can recite SOH-CAH-TOA yet mislabel the hypotenuse the moment the triangle is rotated. This lesson makes them mark the right angle first, then choose the formula.

Objectives — Students can: (1) use sin, cos and tan to find a side or an angle in a right triangle; (2) decide between the sine rule and the cosine rule and state the data each one needs; (3) compute a triangle area with ½ab sinC and explain why it is not base times height.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Pythagoras	3-4-5 triangle hypotenuse	5
Angle unit	calculator in DEG mode	sin30 = 0.5
Roots and ratio	square root of 67	8.185

02 Lesson flow (50 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Label the sides	8-15-17 triangle	sin(theta) = 0.471
8-20 min	Sides and angles	5-12-13	theta = 22.6 deg
20-32 min	Sine rule	A=40, a=12, B=65	b = 16.92
32-44 min	Cosine rule	b=7, c=9, A=60	a = 8.185
44-48 min	Area formula	½ab sinC	27.28
48-50 min	Exit check	three questions collected	independent work

03 Teaching script (read it out)

Scene 1: Mark the hypotenuse before naming ratios

The hypotenuse always faces the right angle and is the longest side. In the 8-15-17 triangle, sin = opposite/hypotenuse = 8/17 = 0.471, cos = 15/17 = 0.882, tan = 8/15 = 0.533, so theta = 28.1 deg.
The usual error is picking the wrong hypotenuse. Circle the right angle, and the side opposite it is the hypotenuse. Also check the calculator is in DEG mode; in RAD every answer is off by a factor of tens.

Student task: circle the right angle on the diagram, label the hypotenuse, then say all three ratios aloud.

Scene 2: Sine rule: one angle and its opposite side are enough

Given A = 40 deg, a = 12 and B = 65 deg: b = a sinB / sinA = 12 x sin65 / sin40 = 16.92.
The rule a/sinA = b/sinB says a side is proportional to the sine of its opposite angle. You need one complete pair before you start; without a pair, switch to the cosine rule. Mention the obtuse case too: the same sine gives two possible angles, so check whether both fit the diagram.

Student task: write the ratio twice, circle the known pair, then substitute.

Scene 3: Cosine rule: three sides, or two sides with the included angle

Given b = 7, c = 9 and included angle A = 60 deg: a squared = 49 + 81 – 2 x 7 x 9 x cos60 deg, so a = 8.185. Area from the same data: ½bc sinA = 27.28.
The cosine rule is Pythagoras with a correction term: at A = 90 deg the cosine term vanishes and it collapses to a squared = b squared + c squared. Students should remember the pairing: cosine for the included angle, sine for the area, both from the same two sides plus the included angle.

Student task: solve for a, then the area, and explain why the area uses sine rather than cosine.

04 Question types and worked steps

Type	Example	Steps	Answer
Find an angle	5-12-13, smallest angle	pick the ratio, then inverse function	22.6 deg
Find a side (sine)	A=40, a=12, B=65	b = a sinB / sinA	16.92
Find a side (cosine)	b=7, c=9, A=60	a squared = b squared + c squared – 2bc cosA	8.185
Area	b=7, c=9, A=60	½bc sinA	27.28

Sanity check (teach this): a = 8.185 must exceed |c – b| = 2 and stay below c + b = 16, and it does. Both angles 28.1 deg and 22.6 deg are under 45 deg because the opposite side is the shorter one; an answer above 60 deg means the ratio was inverted.

05 Independent practice (3 questions)

Question	Answer	What to watch
3-4-5 triangle, find the smallest angle.	36.9 deg	does the shortest side face the smallest angle
A=50, a=10, B=70, find b.	b = 12.27	is there a complete angle-side pair
b=5, c=8, A=120, find a.	a = 11.36	is cos120 taken as negative

06 Common mistakes and what to say

Mistake	Student writes	Say this
Wrong hypotenuse	treats the longest side as adjacent	“The hypotenuse faces the right angle; circle it first.”
Calculator left in RAD	sin30 returns a negative number	“Switch to DEG before anything else in the exam.”
Sine rule inverted twice	multiplies by a again after rearranging	“Side on top, opposite angle below; write it once.”
Missing the obtuse solution	stops after one angle	“The same sine gives two angles; check both when there is no diagram.”

07 Homework (paper, section and question numbers)

Secondary maths chapter index (quadratics, logarithms, trigonometry, probability) — finish trigonometry before logarithms
Secondary prep guide hub — all subjects

08 Exit check (last 5 minutes)

Question	Answer	Decision
5-12-13 triangle, find the smallest angle.	22.6 deg	wrong: redo act one labelling
A=45, a=8, B=60, find b.	9.80	wrong: redo sine rule pairing
b=6, c=10, A=90, find a.	11.66	two wrong: return to act three

09 Tutor FAQ

Which year is this for?

S4 to S6; it covers the compulsory trigonometry and area questions in HKDSE maths.

How many formulas must be memorised?

Three: the ratios, the sine rule and the cosine rule. The area formula follows from the sine rule.

Is there practice attached?

The homework list links to the maths chapter index, pairing with circle geometry and vectors.

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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.