Overview
What you’ll learn
- Name the hypotenuse, opposite and adjacent sides of a right triangle relative to a chosen acute angle.
- State the tangent, sine and cosine ratios and recall SOH-CAH-TOA.
- Use a trigonometric ratio to find an unknown side, given an acute angle and one side.
- Use inverse trigonometric functions to find an unknown angle, given two sides.
- Choose the correct ratio for a given problem and judge whether an answer is sensible.
- Apply trigonometric ratios in real-world right-triangle problems.
In 30 seconds
- 1Relative to an acute angle θ: the hypotenuse faces the right angle, the opposite side faces θ, and the adjacent side touches θ.
- 2The three ratios: tan θ = opp/adj, sin θ = opp/hyp, cos θ = adj/hyp — remembered as SOH-CAH-TOA.
- 3The ratios depend only on the angle, not on the triangle’s size: every 30° right triangle gives the same tan 30°.
- 4Finding a side: pick the ratio linking what you know to what you want, then multiply or divide.
- 5Finding an angle: form the ratio from two sides, then press the inverse button tan⁻¹, sin⁻¹ or cos⁻¹.
- 6Sanity check: the hypotenuse must stay the longest side, and acute angles live between 0° and 90°.
Notes · 1 of 6
Naming the sides
Opposite and adjacent are decided by the angle you care about.
In a right triangle, pick one of the two acute angles and call it θ. The side directly across from θ is the opposite side. The side that touches θ (and is not the hypotenuse) is the adjacent side. The hypotenuse — always the longest side — faces the right angle and touches both acute angles.
| Side | How to spot it |
|---|---|
| Hypotenuse | opposite the right angle; the longest side |
| Opposite | directly across from θ, not touching it |
| Adjacent | one of θ’s own two sides; not the hypotenuse |
Here is the twist: swapping to the other acute angle swaps the labels. The side that was opposite becomes adjacent. Nothing about the triangle changed — only your point of view. Always mark θ and its two labels before touching a formula.
Notes · 2 of 6
The three ratios
Each ratio fixes one side-pairing, and SOH-CAH-TOA keeps them straight.
| Name | Definition | Memory line |
|---|---|---|
| tangent (tan) | tan θ = opposite ÷ adjacent | TOA |
| sine (sin) | sin θ = opposite ÷ hypotenuse | SOH |
| cosine (cos) | cos θ = adjacent ÷ hypotenuse | CAH |
| Ratio | Working | Value |
|---|---|---|
| tan θ | 3 ÷ 4 | 0.75 |
| sin θ | 3 ÷ 5 | 0.6 |
| cos θ | 4 ÷ 5 | 0.8 |
A key fact: the ratio depends only on the angle. Enlarge the triangle and every side doubles — but 6 ÷ 8 is still 0.75. That is why a calculator button labelled tan can work for all triangles in the world at once.
Notes · 3 of 6
Finding a side
One angle, one side, the right ratio — the third side follows.
- 11. Mark θ and label
Opposite, adjacent, hypotenuse — relative to the given angle.
- 22. Choose the ratio
The ratio that contains both your known side and your unknown.
- 33. Write the equation
For example: sin 40° = x/10.
- 44. Solve
Multiply through when the unknown is on top; divide when it sits in a denominator.
| Unknown position | Pattern | Example |
|---|---|---|
| Top (opposite or adjacent) | x = known × ratio | x = 10 × sin 40° ≈ 6.4 |
| Bottom (hypotenuse or adjacent) | x = known ÷ ratio | x = 6 ÷ cos 40° ≈ 7.8 |
Notes · 4 of 6
Finding an angle
Two sides in, one angle out — through the inverse buttons.
When two sides are known and the angle is missing, form the matching ratio and undo it with the inverse function: tan⁻¹, sin⁻¹ or cos⁻¹ (shown as atan, asin or acos on some calculators). For opposite 9 and adjacent 12: tan θ = 9/12 = 0.75, so θ = tan⁻¹(0.75) ≈ 36.9°.
| Known sides | Form | Angle from |
|---|---|---|
| opposite and adjacent | opp ÷ adj | tan⁻¹ |
| opposite and hypotenuse | opp ÷ hyp | sin⁻¹ |
| adjacent and hypotenuse | adj ÷ hyp | cos⁻¹ |
Notes · 5 of 6
Choosing the right ratio
One decision rule: the ratio that knows what you know.
| You know | You want | Use |
|---|---|---|
| angle + opposite | adjacent | tan (divide) |
| angle + adjacent | opposite | tan (multiply) |
| angle + hypotenuse | opposite | sin |
| angle + hypotenuse | adjacent | cos |
| opposite + adjacent | the angle | tan⁻¹ |
| opposite + hypotenuse | the angle | sin⁻¹ |
| adjacent + hypotenuse | the angle | cos⁻¹ |
Notice the pattern: the ratio that contains both your known quantities and your unknown is the only one you can actually compute. If the sides in hand are the opposite and the hypotenuse, tangent is useless — it never mentions the hypotenuse.
Notes · 6 of 6
Trigonometry in the real world
Ladders, ramps, roofs and shadows are all right triangles in disguise.
| Story | Right angle | Hypotenuse |
|---|---|---|
| Ladder against a wall | wall meets ground | the ladder |
| Ramp to a platform | platform meets ground | the ramp |
| Rafter to the roof peak | rafter meets the horizontal ceiling line | the rafter |
| Pole and its shadow | pole meets ground | the sun’s line to the tip (imaginary) |
The routine never changes: sketch, mark the right angle and θ, label the sides, choose the ratio, solve, sanity-check. The sanity check matters most in applications — a 90° ladder angle or a hypotenuse shorter than a leg means the labels slipped.
Revise
Key terms
The words examiners expect you to use precisely.
- Opposite side
- The side directly across from a chosen angle, not touching it.
- Adjacent side
- The side of a chosen angle that is not the hypotenuse.
- Hypotenuse
- The longest side of a right triangle, opposite the right angle.
- Tangent (tan)
- The ratio opposite ÷ adjacent for an acute angle.
- Sine (sin)
- The ratio opposite ÷ hypotenuse for an acute angle.
- Cosine (cos)
- The ratio adjacent ÷ hypotenuse for an acute angle.
Show 6 more key termsShow fewer key terms
- SOH-CAH-TOA
- A memory line for the three ratio definitions.
- Inverse function (tan⁻¹, sin⁻¹, cos⁻¹)
- The calculator operation that returns the angle from a ratio.
- Acute angle
- An angle between 0° and 90°.
- Angle of the ramp
- The angle a sloping object makes with the horizontal ground.
- Ratio
- A comparison of two sides by division; fixed by the angle alone.
- Rounding to 1 decimal place
- The usual precision for angle and length answers in applications, as instructed by the question.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Labelling opposite and adjacent from the wrong angle.
The labels depend entirely on which acute angle you have marked.
Instead: Mark θ first, then read the sides from θ’s point of view.
Mistake: Mixing up sine and tangent, e.g. tan θ = opp/hyp.
The memory lines have collided.
Instead: SOH-CAH-TOA: tangent never involves the hypotenuse.
Show 8 more mistakesShow fewer mistakes
Mistake: Dividing when the unknown is on top (x = 10 sin 40° becomes 10 ÷ sin 40°).
The equation sin 40° = x/10 solves by multiplying both sides by 10.
Instead: Unknown on top → multiply; unknown underneath → divide.
Mistake: Using tan to find an angle from opposite and hypotenuse.
Tangent is defined from opposite and adjacent — the hypotenuse is not in its formula.
Instead: Opposite and hypotenuse → sin θ, so θ = sin⁻¹(opp/hyp).
Mistake: Pressing sin instead of sin⁻¹ when finding an angle.
sin(0.75) returns a ratio-like number, not an angle.
Instead: Angles come from the inverse buttons: tan⁻¹, sin⁻¹, cos⁻¹.
Mistake: Accepting a hypotenuse shorter than a leg in the answer.
The hypotenuse is always the longest side of a right triangle.
Instead: If your computed “hypotenuse” is smaller, the ratio or the labelling is wrong.
Mistake: Rounding the angle to a whole degree and carrying it into the next part.
Early rounding drifts later answers away from the mark-scheme value.
Instead: Keep several decimal places in the working; round only the final answer as instructed.
Mistake: Working trig problems with the calculator in radian mode.
At this level every angle is in degrees; radian mode silently returns wrong values for the same keystrokes.
Instead: Check the DEG indicator before the first button press.
Mistake: Assuming the adjacent side is always the flat bottom side.
“Adjacent” means the leg that touches θ — in a ladder problem it can be the vertical wall.
Instead: Mark θ first, then name the two legs by their relationship to θ.
Mistake: Measuring a ladder’s angle at the wall instead of at the ground.
The angle with the ground lives at the foot of the ladder; the corner at the wall holds its complement.
Instead: Mark the angle in the corner the question names — swapping the corners swaps sin for cos.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Name the sides
Relative to a chosen acute angle θ, the side directly across from it is called
Ready for more practice on Chapter 8: Trigonometric Ratios?
Generate a fresh 5-question drill built from this chapter’s key points. Free with your study account.
Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 5Foundation
- 6Standard
- 4Challenging
15 questions worth 59 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 8 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 8.
Mark scheme
PROChapter 8 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 2Coming soonWeighted Assessment 2 (WA2)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 2 Math Chapter 8 about?
Chapter 8 covers the tangent, sine and cosine of acute angles in right-angled triangles: naming opposite, adjacent and hypotenuse, the SOH-CAH-TOA definitions, finding unknown sides and angles, and simple applications such as ladders, ramps and shadows. (G2 classes usually meet this topic in Sec 3.)
How do I remember the three ratios?
SOH-CAH-TOA: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. Tangent is the only one that never mentions the hypotenuse.
When do I use tan⁻¹, sin⁻¹ or cos⁻¹?
When two sides are known and the angle is missing. Form the ratio that matches your pair of sides — tangent for opposite and adjacent, sine for opposite and hypotenuse, cosine for adjacent and hypotenuse — then press the matching inverse button.
How can I check a trigonometry answer?
Two quick checks: the hypotenuse must remain the longest side, and the angle must sit between 0° and 90°. A ramp-angle question should also pass the common-sense test — real ramps rise only a few degrees.
Up next
Chapter 9: Volume and Surface Area of Pyramids, Cones and Spheres