What is Right Angled Trigonometry in the context of the Cambridge IGCSE Mathematics curriculum?

Right Angled Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of right-angled triangles. In the Cambridge IGCSE Mathematics curriculum, students learn to apply trigonometric ratios such as sine, cosine, and tangent to solve problems involving right-angled triangles.

How do you use trigonometric ratios to find unknown sides in a right-angled triangle?

To find unknown sides in a right-angled triangle using trigonometric ratios, identify the sides relative to the angle of interest: opposite, adjacent, and hypotenuse. Use the sine ratio (opposite/hypotenuse), cosine ratio (adjacent/hypotenuse), or tangent ratio (opposite/adjacent) depending on the given information. Solve the equation to find the unknown side.

What are the common trigonometric ratios used in Right Angled Trigonometry?

The common trigonometric ratios used in Right Angled Trigonometry are sine (sin), cosine (cos), and tangent (tan). These ratios relate the angles of a right-angled triangle to the lengths of its sides: sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

How can you determine the angle of a right-angled triangle using trigonometric ratios?

To determine the angle of a right-angled triangle using trigonometric ratios, use the inverse functions of sine, cosine, or tangent. For example, if you know the lengths of the opposite and hypotenuse sides, use the inverse sine function (sin⁻¹) to find the angle. Similarly, use cos⁻¹ or tan⁻¹ for cosine and tangent ratios, respectively.

What is the importance of the Pythagorean theorem in Right Angled Trigonometry?

The Pythagorean theorem is crucial in Right Angled Trigonometry as it provides a relationship between the sides of a right-angled triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is often used alongside trigonometric ratios to solve problems involving right-angled triangles in the Cambridge IGCSE Mathematics curriculum.

Why is "Calculator in radian mode" a common mistake in Right Angled Trigonometry?

Why it happens: Default mode varies by calculator. How to avoid it: Switch to DEG at the start of the paper. Spot it: 30 -0.99 in radians, 0.5 in degrees. Source: 0580/42 — recurring

Why is "Mislabelling opposite vs adjacent" a common mistake in Right Angled Trigonometry?

Why it happens: Labels depend on which angle you're working from. How to avoid it: Stand at the angle. Hypotenuse is across from the right angle. Opposite is across from the angle. Adjacent is the remaining one.

Why is "Forgetting to apply $\sin^{-1}$ when finding an angle" a common mistake in Right Angled Trigonometry?

Why it happens: Stopping at = 0.42 and writing = 0.42. How to avoid it: If you have = value, your last step is = ^-1(value).

Why is "Rounding intermediate trig values to 2 d.p." a common mistake in Right Angled Trigonometry?

Why it happens: Looks tidier. How to avoid it: Keep at least 4 d.p. (or use ans/memory) until the final answer. Round once at the end.

TrigonometryCambridge IGCSE Mathematics (0580)

Right Angled Trigonometry

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

PreviousNext

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Page 1 / 0

Detailed Study Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Right Angled Trigonometry, ready to print or save offline.

Right-angled Trigonometry — Cambridge IGCSE 0580 Maths Extended (2026)

SOH CAH TOA. Sine, cosine and tangent ratios in right-angled triangles. Find missing sides or angles by labelling Hypotenuse, Opposite, Adjacent first.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E8.1 — Apply Pythagoras' theorem and sine, cosine and tangent ratios for acute angles to the calculation of a side or of an angle of a right-angled triangle.

SOH CAH TOA — the three ratios

Label hyp/opp/adj relative to the angle. Pick the ratio with TWO sides you have or want.

In a right-angled triangle with one acute angle $\theta$ :

$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}.$

Labelling the sides.

Hypotenuse: opposite the right angle (always the longest).
Opposite: across from $\theta$ .
Adjacent: next to $\theta$ AND the right angle.

SOH CAH TOA.

Sin = Opposite over Hypotenuse.
Cos = Adjacent over Hypotenuse.
Tan = Opposite over Adjacent.

Worked. Right-angled triangle, $\theta = 35\degree$ , hypotenuse $= 12\,\text{cm}$ . Find the opposite side.

We have hyp, want opp → use sine.
$\sin 35\degree = \dfrac{\text{opp}}{12} \Rightarrow \text{opp} = 12 \sin 35\degree \approx 6.88\,\text{cm}$ .

Label hyp / opp / adj first — relative to $\theta$ .
Hypotenuse = longest = opposite right angle.
Pick the ratio with the TWO sides involved.
Set calculator to DEGREE mode.

Finding a missing side

Substitute the angle and known side, then rearrange. Multiply if the unknown is in the numerator; divide if in the denominator.

Strategy.

Identify the angle and the two sides (one known, one unknown).
Pick the right ratio (sine, cosine, or tangent).
Substitute, then rearrange to isolate the unknown.

Worked 1 (unknown in numerator). Find $x$ when $\theta = 50\degree$ and adjacent $= 8\,\text{cm}$ , $x$ is opposite.

$\tan 50\degree = \dfrac{x}{8} \Rightarrow x = 8 \tan 50\degree \approx 9.53\,\text{cm}$ .

Worked 2 (unknown in denominator). Find $x$ when $\theta = 40\degree$ and opposite $= 7\,\text{cm}$ , $x$ is hypotenuse.

$\sin 40\degree = \dfrac{7}{x} \Rightarrow x = \dfrac{7}{\sin 40\degree} \approx 10.89\,\text{cm}$ .

Tip. Cross-multiply mentally: when the unknown is on top, multiply both sides by the bottom; when it's on the bottom, swap it with the trig value.

Pick the ratio matching the known and unknown sides.
Unknown on top: multiply.
Unknown on bottom: divide.
Round to 2-3 d.p. or 3 s.f.

Finding a missing angle

Use the inverse trig function: $\sin^{-1}, \cos^{-1}, \tan^{-1}$ .

When two sides are known and you need the angle, use the inverse trig function.

Worked. Right-angled triangle, opposite $= 5\,\text{cm}$ , hypotenuse $= 13\,\text{cm}$ . Find $\theta$ .

$\sin\theta = \dfrac{5}{13} \Rightarrow \theta = \sin^{-1}\left(\dfrac{5}{13}\right) \approx 22.6\degree$ .

Notation. $\sin^{-1} x$ means "the angle whose sine is $x$ " — NOT $\dfrac{1}{\sin x}$ . On most calculators, press SHIFT (or 2nd) then sin.

Common ratios to recognise.

$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$
$0\degree$	0	1	0
$30\degree$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{3}}$
$45\degree$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	$1$
$60\degree$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$
$90\degree$	1	0	undef

Inverse trig: $\sin^{-1}, \cos^{-1}, \tan^{-1}$ .
$\sin^{-1} x \ne \dfrac{1}{\sin x}$ .
Memorise exact values for $30\degree, 45\degree, 60\degree$ .

Angles of elevation and depression

Elevation: looking UP from horizontal. Depression: looking DOWN.

Angle of elevation. From an observer's eye, the angle ABOVE the horizontal up to a target.

Angle of depression. From an observer's eye, the angle BELOW the horizontal down to a target.

Worked. From the top of a $20\,\text{m}$ tower, the angle of depression to a car on the ground is $35\degree$ . How far is the car from the tower's base?

Draw the right-angled triangle: tower (vertical) = $20$ , distance to car (horizontal) = $x$ , angle of depression at top = $35\degree$ .
The angle of depression at the top equals the angle of elevation at the car (alternate angles, parallel horizontals).
$\tan 35\degree = \dfrac{20}{x} \Rightarrow x = \dfrac{20}{\tan 35\degree} \approx 28.6\,\text{m}$ .

Tip. Always draw a clear right-angled triangle and label the angle, opposite, adjacent. Cambridge marks the diagram set-up — even rough work is worth checking.

Elevation: angle UP from horizontal.
Depression: angle DOWN from horizontal.
Depression at top = elevation at base (parallel horizontals).
Always draw and label the triangle.

How it’s examined

Right-angled trig appears every Paper 2 (3-4 marks) and Paper 4 (typically as a sub-question of a longer problem worth 4-6 marks). Examiner reports flag two errors: (i) calculator in radian mode, (ii) using the wrong side label (treating the side OPPOSITE the right angle as adjacent).

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E8.1); 0580/22 May/Jun 2024 — Q9 (right-angled trig); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Right Angled Trigonometry, ready to print or save as PDF.

Step-by-step worked examples — Right Angled Trigonometry

Step-by-step solutions to past-paper-style questions on right angled trigonometry, written exactly the way a tutor would explain them at the board.

1Find an unknown side using cosine
Core• cosine
Question
In a right-angled triangle, the hypotenuse is $10$ cm and one angle is $35\degree$ . Find the side adjacent to that angle.
Step-by-step solution
1. Step 1
  $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$ .
  $\cos 35\degree = \dfrac{x}{10}$
2. Step 2
  Rearrange and compute.
  $x = 10 \cos 35\degree \approx 8.19\,\text{cm}$
Answer
$x \approx 8.19\,\text{cm}$
2Find an unknown angle
Core• Adapted from 0580/22 May/Jun 2024 Q15• inverse trig
Question
A right-angled triangle has opposite side $7$ cm and adjacent side $5$ cm. Find the angle.
Step-by-step solution
1. Step 1
  Use $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$ .
  $\tan\theta = \dfrac{7}{5}$
2. Step 2
  Apply $\tan^{-1}$ .
  $\theta = \tan^{-1}(1.4) \approx 54.5\degree$
Answer
$\theta \approx 54.5\degree$
3Combine Pythagoras and trigonometry
Extended• pythagoras
Question
A ladder of length $6$ m leans against a wall. Its foot is $2$ m from the base of the wall. Find the angle the ladder makes with the ground.
Step-by-step solution
1. Step 1
  Hypotenuse = $6$ , adjacent (along ground) = $2$ .
  $\cos\theta = \dfrac{2}{6} = \dfrac{1}{3}$
2. Step 2
  Compute.
  $\theta = \cos^{-1}\left(\dfrac{1}{3}\right) \approx 70.5\degree$
Answer
$\theta \approx 70.5\degree$
4Angle of elevation
Extended• elevation
Question
From a point $50$ m from the base of a tower, the angle of elevation of the top is $32\degree$ . Find the height of the tower.
Step-by-step solution
1. Step 1
  Horizontal = adjacent, height = opposite. Use $\tan$ .
  $\tan 32\degree = \dfrac{h}{50}$
2. Step 2
  Solve for $h$ .
  $h = 50\tan 32\degree \approx 31.2\,\text{m}$
Answer
$h \approx 31.2\,\text{m}$

Key Formulae — Right Angled Trigonometry

The formulae you need to memorise for right angled trigonometry on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Sine ratio
$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$
When to use
Opposite or hypotenuse known/required.
Cosine ratio
$\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$
When to use
Adjacent or hypotenuse known/required.
Tangent ratio
$\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$
When to use
When both legs are involved (no hypotenuse).
Inverse trigonometric functions
$\theta = \sin^{-1}(x),\ \cos^{-1}(x),\ \tan^{-1}(x)$
When to use
Use to recover angle from a known ratio.

Key Definitions and Keywords — Right Angled Trigonometry

Definitions to memorise and the exact keywords mark schemes credit for right angled trigonometry answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Hypotenuse
Examiner keyword
The longest side of a right-angled triangle, opposite the right angle.
Opposite
Examiner keyword
Side directly across from the angle in question.
Adjacent
Examiner keyword
The leg next to the angle in question (not the hypotenuse).
Angle of elevation
Examiner keyword
The angle measured from the horizontal upward to a higher point.
Angle of depression
Examiner keyword
The angle measured from the horizontal downward to a lower point.

Common Mistakes and Misconceptions — Right Angled Trigonometry

The traps other students keep falling into on right angled trigonometry questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Calculator in radian mode
0580/42 — recurring
Why it happens
Default mode varies by calculator.
How to avoid it
Switch to DEG at the start of the paper. Spot it: $\sin 30 \approx -0.99$ in radians, $0.5$ in degrees.
✕Mislabelling opposite vs adjacent
Why it happens
Labels depend on which angle you're working from.
How to avoid it
Stand at the angle. Hypotenuse is across from the right angle. Opposite is across from the angle. Adjacent is the remaining one.
✕Forgetting to apply $\sin^{-1}$ when finding an angle
Why it happens
Stopping at $\sin\theta = 0.42$ and writing $\theta = 0.42$ .
How to avoid it
If you have $\sin\theta =$ value, your last step is $\theta = \sin^{-1}(\text{value})$ .
✕Rounding intermediate trig values to 2 d.p.
Why it happens
Looks tidier.
How to avoid it
Keep at least 4 d.p. (or use $\text{ans}$ /memory) until the final answer. Round once at the end.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Right Angled Trigonometry — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Right Angled Trigonometry

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Find an unknown side using cosine

2Find an unknown angle

3Combine Pythagoras and trigonometry

4Angle of elevation

Sine ratio

Cosine ratio

Tangent ratio

Inverse trigonometric functions

Hypotenuse

Opposite

Adjacent

Angle of elevation

Angle of depression

✕Calculator in radian mode

✕Mislabelling opposite vs adjacent

✕Forgetting to apply sin⁡−1\sin^{-1}sin−1 when finding an angle

✕Rounding intermediate trig values to 2 d.p.

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Right Angled Trigonometry — frequently asked questions

✕Forgetting to apply $\sin^{-1}$ when finding an angle