Trigonometry

Q: What is the definition of a radian?

A radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius.

Q: How do you convert between degrees and radians?

To convert degrees to radians, multiply by $\frac{\pi}{180}$; to convert radians to degrees, multiply by $\frac{180}{\pi}$.

Q: What are the formulae for arc length and sector area when using radians?

Arc length $ s = r\theta $ and Sector Area $ A = \frac{1}{2}r^2\theta $, where $ \theta $ is in radians.

Q: What are the period and range of the basic sine and cosine functions?

Both $ y = \sin x $ and $ y = \cos x $ have a period of $ 360^\circ $ (or $ 2\pi $ rad) and a range of $ -1 \le y \le 1 $.

Q: What are the period and range of the tangent function, $ y = \tan x $?

The period is $ 180^\circ $ (or $ \pi $ rad) and the range is all real numbers ($ y \in \mathbb{R} $).

Q: How does the transformation $ y = a \sin x $ affect the trigonometric graph?

It results in a vertical stretch by scale factor $ a $, changing the amplitude to $ |a| $ and the range to $ -|a| \le y ≤ |a| $.

Q: How does the transformation $ y = \sin(bx) $ affect the period of the graph?

It results in a horizontal stretch (compression) by factor $ \frac{1}{b} $, changing the period to $ \frac{360^\circ}{b} $ or $ \frac{2\pi}{b} $.

Q: What are the exact values of $ \sin $, $ \cos $, and $ \tan $ for $ 45^\circ $?

$ \sin 45^\circ = \frac{1}{\sqrt{2}} $, $ \cos 45^\circ = \frac{1}{\sqrt{2}} $, and $ \tan 45^\circ = 1 $.

Q: What is the Quotient Identity in trigonometry?

$ \tan \theta = \frac{\sin \theta}{\cos \theta} $.

Q: What is the Pythagorean Identity in trigonometry?

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

Practice

Mathematics (9709) · Cambridge International A Level · 10 min read · Updated: 29 Aug 2026

AS & A Level Mathematics (9709) Pure Mathematics 1: Trigonometry

Hello future mathematicians! Welcome to the exciting world of Trigonometry. This chapter builds on your basic knowledge of SOH CAH TOA and extends it dramatically, allowing you to deal with angles of any size, graph trigonometric functions, and solve powerful cyclic equations.

Trigonometry is vital because it describes all phenomena that repeat cyclically, like waves, sound, light, and orbits. Mastering this section is fundamental for success in Paper 1 and lays the groundwork for advanced identities and calculus in Papers 2 and 3. Let's dive in!

1. Circular Measure: The Radian System (Review of 1.4)

Before we tackle advanced trigonometry, we must ensure you are comfortable using radians, as they are the standard unit for calculus and advanced mathematical work.

What is a Radian?

A radian is simply the angle subtended at the centre of a circle by an arc whose length is equal to the radius.
Analogy: Imagine a piece of string exactly the length of the circle's radius. The angle formed when you wrap that string around the edge is exactly 1 radian.

Conversion Between Degrees and Radians

The core relationship you must memorise is: $$ 180^\circ = \pi \text{ radians} $$

To convert Degrees to Radians: Multiply by $\frac{\pi}{180}$.
To convert Radians to Degrees: Multiply by $\frac{180}{\pi}$.

Quick Review: Key Radian Equivalents

$360^\circ = 2\pi$
$90^\circ = \frac{\pi}{2}$
$45^\circ = \frac{\pi}{4}$
$30^\circ = \frac{\pi}{6}$
$60^\circ = \frac{\pi}{3}$

Formulae for Arc Length and Sector Area

These formulae only work if the angle $\theta$ is in radians. Always check your units!

Arc Length (s): The distance around the curve.
$$ s = r\theta $$
Sector Area (A): The area of the "pizza slice".
$$ A = \frac{1}{2}r^2\theta $$

Key Takeaway (Circular Measure): Always convert to radians if you are asked to use arc length or sector area formulae, or if the question involves calculus (differentiation/integration, which often use angles in radians by default).

2. The Core Trigonometric Graphs and Transformations

You need to be able to sketch and use the graphs of the three basic trigonometric functions for angles of any size. Understanding the shape, period, range, and amplitude is critical, especially when solving equations.

The Sine Function: $y = \sin x$

Period: $360^\circ$ or $2\pi$ radians. (It repeats every $360^\circ$).
Range: $-1 \le y \le 1$.
Amplitude: 1.
Starting Point: Starts at (0, 0). (SINE starts at Zero)

The Cosine Function: $y = \cos x$

Period: $360^\circ$ or $2\pi$ radians.
Range: $-1 \le y \le 1$.
Amplitude: 1.
Starting Point: Starts at (0, 1). (COSINE starts at One)

The Tangent Function: $y = \tan x$

Period: $180^\circ$ or $\pi$ radians. (It repeats faster!)
Range: $y \in \mathbb{R}$ (all real numbers).
Asymptotes: Occur where $\cos x = 0$, i.e., $x = 90^\circ, 270^\circ$, etc.

Graph Transformations (Syllabus Requirement)

You must be able to sketch graphs showing the effect of transformations. These follow the same rules as general function transformations (Section 1.2).

1. Vertical Stretch / Amplitude Change: $y = a \cdot \sin x$

The graph is stretched vertically by factor $a$. The new amplitude is $|a|$ and the range is $-|a| \le y \le |a|$.
Example: For $y = 3 \sin x$, the amplitude is 3, and the range is $-3 \le y \le 3$.

2. Period Change / Horizontal Stretch: $y = \sin (b x)$

The graph is compressed horizontally. The period changes to $\frac{360^\circ}{b}$ or $\frac{2\pi}{b}$. $b$ represents the number of cycles completed in $360^\circ$.
Example: For $y = \cos (2x)$, the period is $180^\circ$ ($\frac{360^\circ}{2}$).

3. Vertical Shift / Midline Change: $y = \sin x + d$

The graph is shifted up by $d$ units. The new midline is $y=d$.
Example: For $y = 1 - \cos 2x$, the whole cosine wave (which usually runs -1 to 1) is flipped ($-\cos$) and shifted up 1 unit. The range is now $0 \le y \le 2$.

4. Horizontal Shift / Phase Change: $y = \sin (x - c)$

The graph is shifted horizontally by $c$ units. (Remember: $-c$ shifts right, $+c$ shifts left).
Example: For $y = \tan(x + \frac{1}{4}\pi)$, the tangent graph is shifted $\frac{\pi}{4}$ units to the left.

Tip for Struggling Students: Sketching Sequence

When sketching a complex transformation, handle them in this order:

Start with the basic shape ($\sin x, \cos x, \tan x$).
Adjust the period using $b$ (horizontal compression).
Adjust the amplitude using $a$ (vertical stretch).
Apply any shifts (horizontal $c$ then vertical $d$).

Key Takeaway (Graphs): Know the fundamental range and period for sine, cosine, and tangent. Transformations affect these properties predictably. Pay close attention to the range required in the question!

3. Essential Identities and Exact Values

A. Exact Values

You must know the exact values for the angles $30^\circ, 45^\circ, 60^\circ$ (and their radian equivalents) without needing a calculator. These are often tested in proofs or questions requiring solutions in surd form.

Table of Exact Values

| Angle $\theta$ | $30^\circ$ ($\frac{\pi}{6}$) | $45^\circ$ ($\frac{\pi}{4}$) | $60^\circ$ ($\frac{\pi}{3}$) |
|---|---|---|---|
| $\sin \theta$ | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ or $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ |
| $\cos \theta$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ or $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ |
| $\tan \theta$ | $\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$ | $1$ | $\sqrt{3}$ |

Did you know? These values come from just two triangles: an equilateral triangle split in half (for 30° and 60°) and a square split in half (for 45°).

B. The P1 Trigonometric Identities

In P1, you only need to master two core identities. These are your tools for simplifying expressions and converting equations into a single function (often a quadratic).

1. The Quotient Identity

$$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$

2. The Pythagorean Identity

This comes directly from Pythagoras' Theorem applied to a right-angled triangle in the unit circle (where the hypotenuse is 1). $$ \sin^2 \theta + \cos^2 \theta = 1 $$

This identity is incredibly flexible! You should be comfortable rearranging it into its two important forms:

$$ \sin^2 \theta = 1 - \cos^2 \theta $$
$$ \cos^2 \theta = 1 - \sin^2 \theta $$

Common Mistake Alert!

Don't confuse $\sin^2 \theta$ (which means $(\sin \theta)^2$) with $\sin 2\theta$ (which is the double angle formula, NOT in P1!). Also, remember that $\sin \theta$ is NOT the same as $\sin(\theta)$, although they are often used interchangeably.

Key Takeaway (Identities): If you see $\sin$ and $\cos$ together, or a squared term, think about using $\sin^2 \theta + \cos^2 \theta = 1$ to simplify or convert the expression into a quadratic equation.

4. Solving Simple Trigonometric Equations

Solving trig equations means finding the angles, usually within a specified range (e.g., $0^\circ \le x \le 360^\circ$ or $-\pi < x < \pi$), that satisfy the given equation.

The CAST Diagram / All Students Take Calculus

Since trig functions are periodic, there is usually more than one solution within the given interval. The CAST diagram helps you find the angles in all four quadrants that have the same basic ratio.

A (All): All ratios ($\sin, \cos, \tan$) are positive (Quadrant 1: $0^\circ \text{ to } 90^\circ$).
S (Sine): Only $\sin$ is positive (Quadrant 2: $90^\circ \text{ to } 180^\circ$).
T (Tangent): Only $\tan$ is positive (Quadrant 3: $180^\circ \text{ to } 270^\circ$).
C (Cosine): Only $\cos$ is positive (Quadrant 4: $270^\circ \text{ to } 360^\circ$).

How to Find the Second Solution:

If basic angle is $\alpha$:
Quadrant 1: $\alpha$
Quadrant 2: $180^\circ - \alpha$ (or $\pi - \alpha$)
Quadrant 3: $180^\circ + \alpha$ (or $\pi + \alpha$)
Quadrant 4: $360^\circ - \alpha$ (or $2\pi - \alpha$)

Don't worry if this seems tricky at first! Practice makes perfect. Follow the structured method below every time.

Step-by-Step Guide to Solving $3 \cos x + 1 = 0$ for $0^\circ \le x \le 360^\circ$

Step 1: Isolate the Trigonometric Ratio

We need $\cos x$ by itself: $$ 3 \cos x = -1 $$ $$ \cos x = -\frac{1}{3} $$

Step 2: Find the Basic Angle ($\alpha$)

We ignore the negative sign for now and use $\cos^{-1}$ to find the acute angle $\alpha$: $$ \alpha = \cos^{-1}\left(\frac{1}{3}\right) \approx 70.53^\circ $$

Step 3: Use CAST or Graph to Determine Quadrants

Since $\cos x$ is negative, solutions must lie in the S (Sine) and T (Tangent) quadrants (Quadrants 2 and 3).

Step 4: Calculate the Final Solutions

Solution 1 (Quadrant 2): $x = 180^\circ - \alpha = 180^\circ - 70.53^\circ \approx 109.5^\circ$ (1 d.p. for angles).
Solution 2 (Quadrant 3): $x = 180^\circ + \alpha = 180^\circ + 70.53^\circ \approx 250.5^\circ$ (1 d.p.).

Both $109.5^\circ$ and $250.5^\circ$ are within the range $0^\circ \le x \le 360^\circ$.

Handling Complex Angles (e.g., $2x$ or $x+30^\circ$)

If the angle itself is transformed (e.g., $3 \sin(2x) = 1$), the domain must also be transformed first.

Example: Solve $\sin(2x) = 0.5$ for $0^\circ \le x \le 360^\circ$.

Step 1: Transform the Domain

If $0^\circ \le x \le 360^\circ$, then the new domain for $2x$ is $0^\circ \le 2x \le 720^\circ$. We must find all solutions in two full rotations.

Step 2: Find Basic Angle and Solutions for $2x$

Basic angle $\alpha = \sin^{-1}(0.5) = 30^\circ$. Since $\sin$ is positive, we use Quadrants 1 and 2.
Solutions in the first rotation ($0^\circ \text{ to } 360^\circ$):

$2x = 30^\circ$ (Q1)
$2x = 180^\circ - 30^\circ = 150^\circ$ (Q2)

Solutions in the second rotation ($360^\circ \text{ to } 720^\circ$):

$2x = 30^\circ + 360^\circ = 390^\circ$
$2x = 150^\circ + 360^\circ = 510^\circ$

Step 3: Solve for $x$

Divide all angles by 2: $$ x = 15^\circ, 75^\circ, 195^\circ, 255^\circ $$

Solving Trigonometric Quadratic Equations

Many complex trig problems require the use of the identities to turn the equation into a quadratic form (in terms of $\sin \theta$, $\cos \theta$, or $\tan \theta$).

Example: Solve $3 \sin^2 \theta - 5 \cos \theta - 1 = 0$.

Step 1: Convert to a Single Trigonometric Function

Use the identity $\sin^2 \theta = 1 - \cos^2 \theta$: $$ 3(1 - \cos^2 \theta) - 5 \cos \theta - 1 = 0 $$ $$ 3 - 3 \cos^2 \theta - 5 \cos \theta - 1 = 0 $$ $$ 2 - 3 \cos^2 \theta - 5 \cos \theta = 0 $$ Multiply by -1 and rearrange: $$ 3 \cos^2 \theta + 5 \cos \theta - 2 = 0 $$

Step 2: Solve the Quadratic

Let $y = \cos \theta$. The equation is $3y^2 + 5y - 2 = 0$. Factorising gives $(3y - 1)(y + 2) = 0$. Thus, $y = \frac{1}{3}$ or $y = -2$.

So, $\cos \theta = \frac{1}{3}$ or $\cos \theta = -2$.

Step 3: Filter Invalid Solutions and Find Angles

Since the range of $\cos \theta$ is $-1 \le \cos \theta \le 1$, the solution $\cos \theta = -2$ is impossible.
We only solve $\cos \theta = \frac{1}{3}$.

Follow Steps 2–4 from the previous method for $\cos \theta = \frac{1}{3}$.

Quick Review: Inverse Notation Principal Values

The notation $\sin^{-1}x$, $\cos^{-1}x$, and $\tan^{-1}x$ refers to the principal value, which is the single answer your calculator gives you. These ranges are:

$\sin^{-1}x$: $-\frac{\pi}{2} \le y \le \frac{\pi}{2}$ (or $-90^\circ \le y \le 90^\circ$)
$\cos^{-1}x$: $0 \le y \le \pi$ (or $0^\circ \le y \le 180^\circ$)
$\tan^{-1}x$: $-\frac{\pi}{2} < y < \frac{\pi}{2}$ (or $-90^\circ < y < 90^\circ$)

Always use the calculator value (the basic angle, $\alpha$) to generate all required solutions in the specified domain.

Key Takeaway (Solving Equations): Always adjust the interval first if the angle is not just $\theta$. Use identities to create a quadratic form when necessary, and always check that your final angles fall within the initial specified range.

References

Cambridge International AS & A Level Mathematics (9709) Specification & Syllabus Overview — Cambridge University Press & Assessment
Trigonometry in Cambridge International A Level Mathematics (9709) — Tutopiya

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the definition of a radian?

A radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius.

How do you convert between degrees and radians?

To convert degrees to radians, multiply by $\frac{\pi}{180}$; to convert radians to degrees, multiply by $\frac{180}{\pi}$.

What are the formulae for arc length and sector area when using radians?

Arc length $ s = r\theta $ and Sector Area $ A = \frac{1}{2}r^2\theta $, where $ \theta $ is in radians.

What are the period and range of the basic sine and cosine functions?

Both $ y = \sin x $ and $ y = \cos x $ have a period of $ 360^\circ $ (or $ 2\pi $ rad) and a range of $ -1 \le y \le 1 $.

What are the period and range of the tangent function, $ y = \tan x $?

The period is $ 180^\circ $ (or $ \pi $ rad) and the range is all real numbers ($ y \in \mathbb{R} $).

How does the transformation $ y = a \sin x $ affect the trigonometric graph?

It results in a vertical stretch by scale factor $ a $, changing the amplitude to $ |a| $ and the range to $ -|a| \le y ≤ |a| $.

How does the transformation $ y = \sin(bx) $ affect the period of the graph?

It results in a horizontal stretch (compression) by factor $ \frac{1}{b} $, changing the period to $ \frac{360^\circ}{b} $ or $ \frac{2\pi}{b} $.

What are the exact values of $ \sin $, $ \cos $, and $ \tan $ for $ 45^\circ $?

$ \sin 45^\circ = \frac{1}{\sqrt{2}} $, $ \cos 45^\circ = \frac{1}{\sqrt{2}} $, and $ \tan 45^\circ = 1 $.

What is the Quotient Identity in trigonometry?

$ \tan \theta = \frac{\sin \theta}{\cos \theta} $.

What is the Pythagorean Identity in trigonometry?

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

How do you find the second solution for a sine equation using the basic angle $ \alpha $?

If $ \sin x $ is positive, the solutions are in Q1 ($ \alpha $) and Q2 ($ 180^\circ - \alpha $).

What is the first step when solving a trig equation with a transformed angle like $ \sin(2x) = k $?

The first step is to transform the domain (interval) to match the angle in the function (e.g., if $ 0 ≤ x ≤ 360 $, then $ 0 ≤ 2x ≤ 720 $).

How do you solve a trigonometric equation that is in quadratic form?

Use identities to reach a single trig function, substitute (e.g., $ y = \cos \theta $), factorise, solve for the trig ratio, and then find the angles.

What is meant by the "principal value" of an inverse trigonometric function?

The principal value is the single angle value returned by a calculator (e.g., $ -90^\circ ≤ ∂ ≤ 90^\circ $ for sine).

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