What are the basic trigonometric ratios used in Cambridge IGCSE Additional Mathematics?

The basic trigonometric ratios used in Cambridge IGCSE Additional Mathematics are sine (sin), cosine (cos), and tangent (tan). These ratios relate the angles and sides of right-angled triangles. Specifically, 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 do you solve trigonometric equations in the Cambridge IGCSE curriculum?

To solve trigonometric equations in the Cambridge IGCSE curriculum, you first need to isolate the trigonometric function. Then, use inverse trigonometric functions to find the angle that satisfies the equation. It's important to consider all possible solutions within the given domain, often using the unit circle or trigonometric identities to find additional solutions.

What is the significance of the unit circle in trigonometry?

The unit circle is significant in trigonometry because it provides a geometric representation of the trigonometric functions. It is a circle with a radius of one centered at the origin of a coordinate plane. The unit circle allows students to understand how the sine, cosine, and tangent functions relate to angles and coordinates, and it helps in visualizing the periodic nature of these functions.

How are trigonometric identities used in solving problems in Additional Mathematics?

Trigonometric identities are used in Additional Mathematics to simplify and solve complex trigonometric expressions and equations. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities. These identities allow students to transform expressions into more manageable forms, making it easier to solve equations or prove mathematical statements.

What are the applications of trigonometry in real-world scenarios?

Trigonometry has numerous real-world applications, including in fields such as engineering, physics, architecture, and astronomy. It is used to calculate distances, angles, and heights in construction and navigation. In physics, trigonometry helps in understanding wave patterns and oscillations. These applications highlight the importance of trigonometry beyond the classroom, making it a vital component of the Cambridge IGCSE Additional Mathematics curriculum.

Why is "Giving only one solution to a trig equation" a common mistake in Trigonometry?

Why it happens: Calculator gives one value. How to avoid it: Trig equations typically have MULTIPLE solutions in a given range. Use symmetry: x = a → x = a AND x = - a. x = a → x = a AND x = - a (or 2 - a). Source: 0606 Examiner Reports 2022-2024

Why is "Mixing radians and degrees" a common mistake in Trigonometry?

Why it happens: Calculator mode confusion. How to avoid it: Check the calculator's MODE matches the question's units. Range in radians (0 x 2) → use radian mode.

Why is "Working from BOTH sides toward the middle in identity proofs" a common mistake in Trigonometry?

Why it happens: Feels symmetric. How to avoid it: Mark scheme expects you to start from ONE side (usually LHS) and transform until it equals the OTHER side. Don't manipulate both sides simultaneously.

TrigonometryCambridge IGCSE Additional Mathematics (0606)

Trigonometry

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Trigonometry Study Notes — Cambridge IGCSE Additional Mathematics 0606 (2025-2027 syllabus)

Trig functions and their reciprocals. Pythagorean identity and its variants. Solving trig equations across $[0, 2\pi]$ . Sketching trig graphs with amplitude, period, and phase shift.

What you’ll learn

Mapped to the Cambridge IGCSE 0606 syllabus (2025-2027).

1.10.1 — Use the trigonometric and reciprocal trig functions.
1.10.2 — Apply the Pythagorean and tangent identities.
1.10.3 — Solve trig equations in given ranges.
1.10.4 — Sketch trig graphs and their transformations.

Trig functions and identities

Three primary, three reciprocal. One key identity.

Primary functions.

$\sin\theta$ — opposite over hypotenuse.
$\cos\theta$ — adjacent over hypotenuse.
$\tan\theta = \frac{\sin\theta}{\cos\theta}$ — opposite over adjacent.

Reciprocals.

$\sec\theta = \frac{1}{\cos\theta}$ .
$\csc\theta = \frac{1}{\sin\theta}$ .
$\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$ .

Pythagorean identity. $\sin^2\theta + \cos^2\theta = 1$ .

Variants (derived by dividing by $\cos^2$ or $\sin^2$ ):

$1 + \tan^2\theta = \sec^2\theta$ .
$1 + \cot^2\theta = \csc^2\theta$ .

Cambridge tip. Memorise the Pythagorean identity AND the two variants. They appear in proofs and equations frequently.

Three primary + three reciprocal trig functions.
$\sin^2 + \cos^2 = 1$ — most important identity.
Variants for $\sec, \csc, \cot$ .

Solving trig equations

Find one solution; use symmetry and periodicity to find others.

Procedure.

Manipulate to a basic form like $\sin x = a$ , $\cos x = a$ , or $\tan x = a$ .
Find the principal value with $\arcsin$ , $\arccos$ , or $\arctan$ .
Use symmetry to find all solutions in the given range.

Symmetry rules.

$\sin x = a$ → $x = \arcsin a$ AND $x = \pi - \arcsin a$ (and add $2\pi k$ ).
$\cos x = a$ → $x = \arccos a$ AND $x = -\arccos a$ (or $2\pi - \arccos a$ ).
$\tan x = a$ → $x = \arctan a$ (and add $\pi k$ ).

Quadratics in trig. If equation looks like $a\sin^2 + b\sin + c = 0$ , substitute $u = \sin x$ , solve as a regular quadratic, then solve the resulting trig equations for each $u$ value.

Worked walkthrough. $2\sin^2 x - 3\sin x + 1 = 0$ for $0 \le x \le 2\pi$ .

Quadratic in $\sin x$ : $(2\sin x - 1)(\sin x - 1) = 0$ .
$\sin x = \frac{1}{2}$ → $x = \frac{\pi}{6}, \frac{5\pi}{6}$ .
$\sin x = 1$ → $x = \frac{\pi}{2}$ .
All solutions: $\frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}$ .

Cambridge tip. When asked for solutions in a RANGE, give ALL solutions in that range, not just the principal one.

Solve, find principal, apply symmetry.
Quadratic in trig — substitute and factorise.
Don't miss solutions — check the range.

See the full worked example for trigonometry →

Proving identities

Manipulate one side until it matches the other.

Procedure. Start from ONE side (typically LHS). Use:

Pythagorean identity to swap $\sin^2$ for $1 - \cos^2$ (or vice versa).
$\tan = \sin / \cos$ .
Common denominators to combine fractions.
Conjugate-style multiplication when terms like $1 - \cos$ appear.

Continue until LHS matches RHS.

Worked walkthrough. Prove $\frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$ .

LHS $\cdot \frac{1 + \cos\theta}{1 + \cos\theta} = \frac{(1 - \cos\theta)(1 + \cos\theta)}{\sin\theta(1 + \cos\theta)}$ .
Numerator: $1 - \cos^2\theta = \sin^2\theta$ .
$= \frac{\sin^2\theta}{\sin\theta(1 + \cos\theta)} = \frac{\sin\theta}{1 + \cos\theta}$ . ✓

Cambridge tip. Don't manipulate both sides — pick ONE and work toward the other. Mark scheme penalises bidirectional manipulation.

Work from ONE side.
Use Pythagorean identity, tan = sin/cos, common denominators.
Conjugate trick for $1 \pm \cos$ or $1 \pm \sin$ .

See the full worked example for trigonometry →

Trig graphs and transformations

$y = a\sin(bx + c) + d$ — amplitude, period, phase, vertical shift.

Standard graphs.

$y = \sin x$ : amplitude 1, period $2\pi$ , range $[-1, 1]$ .
$y = \cos x$ : amplitude 1, period $2\pi$ , range $[-1, 1]$ . Phase-shifted $\sin$ .
$y = \tan x$ : period $\pi$ , asymptotes at $x = \frac{\pi}{2} + k\pi$ .

General form $y = a\sin(bx + c) + d$ .

$|a|$ = amplitude.
Period = $\frac{2\pi}{|b|}$ .
Phase shift = $-\frac{c}{b}$ (negative if $c > 0$ — shifts left).
$d$ = vertical shift.

Worked walkthrough. $y = 2\sin\left(x + \frac{\pi}{4}\right)$ .

Amplitude 2.
Period $2\pi$ (coefficient of $x$ is 1).
Phase shift: $\frac{\pi}{4}$ to the LEFT.

Cambridge tip. When sketching, mark the key points: maxima, minima, x-intercepts, y-intercept.

Amplitude, period, phase, vertical shift.
$y = a\sin(bx + c) + d$ — general form.
Mark key points on sketches.

See the full worked example for trigonometry →

How it’s examined

Trigonometry is heavy on every Paper 1. Most-tested: solve trig equation (5-7 marks), prove identity (5-7 marks), sketch graph with transformations (4-6 marks). Examiner reports flag missed solutions and wrong calculator mode as common errors.

Sources: Cambridge IGCSE Additional Mathematics 0606 syllabus (2025-2027); 0606 Examiner Reports 2022-2024; 0606/12 Oct/Nov 2024 question paper and mark scheme. Last reviewed 2026-05-09.

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Step-by-step worked examples — Trigonometry

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

1Solve a trigonometric equation (6 marks)
Extended• Adapted from 0606/12 Oct/Nov 2024 Q8• trig-equation
Question
Solve $2\sin^2 x - 3\sin x + 1 = 0$ for $0 \le x \le 2\pi$ . (6 marks)
Step-by-step solution
1. Step 1
  Treat as quadratic in $\sin x$ (2 marks). Let $u = \sin x$ . Then $2u^2 - 3u + 1 = 0$ , so $(2u - 1)(u - 1) = 0$ , giving $u = \frac{1}{2}$ or $u = 1$ .
2. Step 2
  Solve $\sin x = \frac{1}{2}$ (2 marks). $x = \frac{\pi}{6}$ and $x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$ .
3. Step 3
  Solve $\sin x = 1$ (1 mark). $x = \frac{\pi}{2}$ .
4. Step 4
  Combine all solutions (1 mark). $x = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}$ .
Answer
$x = \dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\pi}{6}$ .
2Prove a trig identity (5 marks)
Extended• identity
Question
Prove that $\dfrac{1 - \cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1 + \cos\theta}$ . (5 marks)
Step-by-step solution
1. Step 1
  Multiply LHS by $\frac{1 + \cos\theta}{1 + \cos\theta}$ (1 mark). $\dfrac{(1 - \cos\theta)(1 + \cos\theta)}{\sin\theta(1 + \cos\theta)}$ .
2. Step 2
  Expand numerator (2 marks). $(1 - \cos\theta)(1 + \cos\theta) = 1 - \cos^2\theta = \sin^2\theta$ (using $\sin^2 + \cos^2 = 1$ ).
3. Step 3
  Simplify (2 marks). $\dfrac{\sin^2\theta}{\sin\theta(1 + \cos\theta)} = \dfrac{\sin\theta}{1 + \cos\theta}$ . ✓
Answer
Identity proven via the Pythagorean identity.
3Sketch a transformed trig graph (4 marks)
Extended• graphs
Question
Sketch $y = 2\sin\left(x + \dfrac{\pi}{4}\right)$ for $0 \le x \le 2\pi$ . State the amplitude, period, and phase shift. (4 marks)
Step-by-step solution
1. Step 1
  Amplitude (1 mark). Coefficient of $\sin$ is 2, so amplitude = 2. Maximum value 2; minimum $-2$ .
2. Step 2
  Period (1 mark). $\sin x$ has period $2\pi$ ; the coefficient of $x$ is 1, so period unchanged at $2\pi$ .
3. Step 3
  Phase shift (1 mark). $x + \frac{\pi}{4}$ shifts the graph $\frac{\pi}{4}$ to the LEFT.
4. Step 4
  Sketch (1 mark). Standard $\sin$ shape, amplitude 2, shifted $\frac{\pi}{4}$ left. At $x = 0$ : $y = 2\sin(\pi/4) = \sqrt{2}$ .
Answer
Amplitude 2, period $2\pi$ , phase shift $\pi/4$ left.

Key Formulae — Trigonometry

The formulae you need to memorise for trigonometry on the Cambridge IGCSE 0606 paper, with every variable defined in plain English and a note on when to use it.

Pythagorean identity
$\sin^2\theta + \cos^2\theta = 1$
$\theta$
Any angle
When to use
Converting between $\sin^2$ and $1 - \cos^2$ (or vice versa) when proving identities or solving equations.
Tan identity
$\tan\theta = \frac{\sin\theta}{\cos\theta}$
$\theta$
Angle (with $\cos\theta \ne 0$ )
When to use
Switching between $\tan$ and a sin/cos ratio.
Reciprocal trig functions
$\sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}, \quad \cot\theta = \frac{1}{\tan\theta}$
$\theta$
Angle (with appropriate restrictions)
When to use
Converting between trig and reciprocal forms.
Reciprocal Pythagorean identities
$1 + \tan^2\theta = \sec^2\theta, \quad 1 + \cot^2\theta = \csc^2\theta$
$\theta$
Angle
When to use
Identities involving sec, csc, cot. Derived from sin² + cos² = 1.

Key Definitions and Keywords — Trigonometry

Definitions to memorise and the exact keywords mark schemes credit for trigonometry answers — sharpened from recent examiner reports for the 2026 0606 sitting.

Amplitude
Examiner keyword
Half the difference between the maximum and minimum values of a periodic function. For $y = a\sin x$ , amplitude = $|a|$ .
Period
Examiner keyword
The smallest positive value $p$ such that $f(x + p) = f(x)$ for all $x$ . $\sin$ and $\cos$ have period $2\pi$ ; $\tan$ has period $\pi$ .
Phase shift
Examiner keyword
Horizontal translation of a periodic function. $y = \sin(x + c)$ shifts $\sin x$ by $-c$ along the x-axis.
Asymptote (in trig)
$\tan x$ has vertical asymptotes at $x = \frac{\pi}{2} + k\pi$ (where $\cos x = 0$ ).

Common Mistakes and Misconceptions — Trigonometry

The traps other students keep falling into on trigonometry questions — taken from recent Cambridge IGCSE 0606 examiner reports and mark schemes — and how to avoid them.

✕Giving only one solution to a trig equation
0606 Examiner Reports 2022-2024
Why it happens
Calculator gives one value.
How to avoid it
Trig equations typically have MULTIPLE solutions in a given range. Use symmetry: $\sin x = a$ → $x = \arcsin a$ AND $x = \pi - \arcsin a$ . $\cos x = a$ → $x = \arccos a$ AND $x = -\arccos a$ (or $2\pi - \arccos a$ ).
✕Mixing radians and degrees
Why it happens
Calculator mode confusion.
How to avoid it
Check the calculator's MODE matches the question's units. Range in radians ( $0 \le x \le 2\pi$ ) → use radian mode.
✕Working from BOTH sides toward the middle in identity proofs
Why it happens
Feels symmetric.
How to avoid it
Mark scheme expects you to start from ONE side (usually LHS) and transform until it equals the OTHER side. Don't manipulate both sides simultaneously.

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.

Trigonometry — frequently asked questions

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

Trigonometry

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Solve a trigonometric equation (6 marks)

2Prove a trig identity (5 marks)

3Sketch a transformed trig graph (4 marks)

Pythagorean identity

Tan identity

Reciprocal trig functions

Reciprocal Pythagorean identities

Amplitude

Period

Phase shift

Asymptote (in trig)

✕Giving only one solution to a trig equation

✕Mixing radians and degrees

✕Working from BOTH sides toward the middle in identity proofs

Practice questions

Past paper style quiz

Assess your understanding

Trigonometry — frequently asked questions