S3 Maths | Trigonometric Ratios Lesson Guide for Tutors | Sine, Cosine, Tangent, Sides and Angles

2026-11-10

Tutor lesson-prep guides · Secondary Maths

Students can chant SOH CAH TOA yet freeze when a diagram appears: the hypotenuse gets mislabelled, the calculator sits in RAD mode, and an answer of 1.3 for a sine is copied down without a blink. The fix is the order of work: label first, choose the ratio second.

Objectives — Students can: (1) label the opposite, adjacent and hypotenuse on a right-angled triangle and write the three ratios; (2) find a missing side from one side and one angle; (3) find an angle from two sides using the inverse function, to one decimal place; (4) use the fact that sine and cosine never exceed 1 to catch their own mistakes.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Right-angled triangles	The hypotenuse faces the right angle; opposite and adjacent follow the chosen angle	Points to all three sides
Squares and roots	Three squared plus four squared	Twenty-five, root five
Calculator mode	The screen must show D or DEG	In RAD mode sine of 30 degrees is not 0.5

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Error diagnosis	Ask for the opposite side when the hypotenuse is 5 and the angle 30 degrees	Answers 2.5 or cannot start
5-15 min	Defining the ratios	Model the labelling order: cross the right angle, mark H, then O and A	Labels two versions of one triangle
15-25 min	Sides and angles	Insist on writing the equation before touching the calculator	Writes two equations and evaluates
25-33 min	Special angles	Drill the five values for 30, 45 and 60 degrees	Writes them from memory and checks a partner
33-40 min	Independent practice	Circulates without hints	Completes four questions with units
40-45 min	Exit check	Collects three answers	Answers orally with a reason

03 Teaching script (read it out)

Scene 1: Three ratios: label first, write later

Cross out the right angle, label the hypotenuse H. Pick the angle the question asks about, draw an arrow to the side facing it and label it O, then label the remaining side A. Only then write sin = O/H, cos = A/H, tan = O/A.
Choose the other acute angle in the same triangle and O and A swap, so the labels follow the angle in the question, not the picture.

Students label O, A and H on the same triangle twice, once for each acute angle

Scene 2: One equation, one unknown

Hypotenuse 5 and angle 30 degrees, find the opposite side: sin 30 = O/5, so O = 5 times 0.5 = 2.5.
Opposite 3 and adjacent 4, find the angle: tan = 3/4 = 0.75, and the inverse function gives 36.9 degrees.
Write the equation before pressing anything; pressing sine of 3 means nothing because a sine takes an angle.

Students write both equations; the tutor marks the equations only, not the numbers

Scene 3: Special angles and sanity checks

Memorise five values: sin 30 = 0.5, cos 60 = 0.5, sin 45 is about 0.707, sin 60 is about 0.866, tan 45 = 1.
Three checks: sine and cosine never exceed 1; the hypotenuse is the longest side, so any ratio with the hypotenuse underneath is below 1; a larger angle has a larger sine. An answer of 1.3 always means a wrong label or a wrong mode.

Students apply the three checks to their own answers and circle the doubtful one

04 Question types and worked steps

Type	Example	Steps	Answer
Find the opposite side	Hypotenuse 10, angle 30 degrees	Label, choose sine, write O = 10 sin 30	5
Find an angle	Opposite 3, adjacent 4	Label, choose tangent, use the inverse, one decimal place	36.9 degrees
Special angles	tan 45 and sin 60	Recall the table and write the values	1 and 0.866
Find the third side	Hypotenuse 13, opposite 5	Use Pythagoras for the adjacent side, then find the angle	Adjacent 12, angle 22.6 degrees

Sanity check (teach this): Sanity check: sine and cosine always sit between 0 and 1, so 0.5 for sin 30 is plausible and 1.5 is impossible; the hypotenuse is the longest side, so a hypotenuse of 5 with an opposite of 3 forces an adjacent of 4, and three squared plus four squared gives 25, whose root is 5; tan 45 equals 1 because the two legs are equal, so any tangent above 1 means the opposite side is longer than the adjacent side and the diagram should show that; an angle found from opposite 3 and adjacent 4 must be under 45 degrees, so 36.9 fits and 76 does not.

05 Independent practice (3 questions)

Question	Answer	What to watch
Side	Hypotenuse 10, angle 30 degrees, find the opposite side	5
Angle	Opposite 3, adjacent 4, find the angle	36.9 degrees
Special angle	What is tan 45	1
Side	Hypotenuse 13, opposite 5, find the adjacent side	12

06 Common mistakes and what to say

Mistake	Student writes	Say this
Mislabelled hypotenuse	The longest side written as the opposite side	The hypotenuse always faces the right angle. Cross the right angle first, then write H.
Calculator left in RAD	sin 30 gives minus 0.988	If there is no D on the screen you are in radians. You want 0.5, so check the mode before every question.
A length inside sine	Writing sin 3 = 0.75	A sine always takes an angle. A side over a side is a tangent.
Ignoring an answer above 1	Copying down sin = 1.3	A sine can never pass 1, so the labels or the mode are wrong, not the arithmetic.

07 Homework (paper, section and question numbers)

Secondary maths chapter index — All chapters, starting with the two either side of this one
Teaching guide index — The same ten sections for other subjects
Redo this chapter — Worksheet questions 1 to 8, writing the ratio used for questions 5 to 8

08 Exit check (last 5 minutes)

Question	Answer	Decision
Hypotenuse 10, angle 30 degrees, find the opposite side	5	Answering 8.66 means cosine was used; reteach act one
Angle from opposite 3 and adjacent 4	36.9 degrees	Answering 53.1 means the legs were swapped
Can a sine equal 1.2	No	Answering yes means the sanity check is not yet in place

09 Tutor FAQ

Must the special angles be memorised?

Yes. The five values decide how fast a paper is finished and no table is given. Derive them once from a 3-4-5 triangle and a square diagonal, then drill.

Why write the equation first?

One wrong key ruins the whole answer, and a written equation earns method marks even when the arithmetic slips.

Why one decimal place?

That is the usual S3 requirement unless the question says otherwise. Extra digits add no marks and invite copying errors.

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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.