Trigonometry

Q: What are the definitions of the reciprocal trigonometric functions secant, cosecant, and cotangent?

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

Q: What is the "Co-Rule" memory aid for reciprocal functions?

The reciprocal function starts with "co" if the original does not, and vice versa: Sine $\leftrightarrow$ Cosecant, Cosine $\leftrightarrow$ Secant, and Tangent $\leftrightarrow$ Cotangent.

Q: Where do the asymptotes occur on the graphs of reciprocal trigonometric functions?

Asymptotes occur wherever the original function equals zero, as division by zero is undefined (e.g., $\sec \theta$ has asymptotes where $\cos \theta = 0$).

Q: What is the general strategy for solving equations involving reciprocal trigonometric functions?

Rewrite the reciprocal function as its primary counterpart (sine, cosine, or tangent) and solve the resulting equation.

Q: What is the Pythagorean identity relating secant and tangent?

$\sec^2 \theta = 1 + \tan^2 \theta$

Q: What is the Pythagorean identity relating cosecant and cotangent?

$\csc^2 \theta = 1 + \cot^2 \theta$

Q: What are the compound angle formulae for sine and cosine?

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ and $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$.

Q: What is the compound angle formula for tangent?

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Q: What are the three different forms of the cosine double angle formula?

1) $\cos 2A = \cos^2 A - \sin^2 A$ 2) $\cos 2A = 2 \cos^2 A - 1$ 3) $\cos 2A = 1 - 2 \sin^2 A$

Q: What is the double angle formula for tangent?

$\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}$

Practice

Mathematics (9709) · Cambridge International A Level · 9 min read · Updated: 29 Aug 2026

Pure Mathematics 2 (Paper 2) Study Notes: Trigonometry

Hello future Mathematician! Welcome to the Trigonometry chapter for Paper 2. If you mastered the basics of $\sin$, $\cos$, and $\tan$ in Pure Mathematics 1 (P1), then P2 is where we unlock their super powers!

In this chapter, we introduce three new functions and a wealth of powerful identities that will allow us to simplify complex expressions and solve much trickier trigonometric equations. Don't worry if this seems like a lot of formulae; we’ll give you tricks to remember them and show you exactly when and why to use them.

Key Takeaway from P1 (Prerequisite Knowledge)

Before diving in, make sure you are comfortable with:

The graphs of $y = \sin x$, $y = \cos x$, and $y = \tan x$.
The identity $\sin^2 \theta + \cos^2 \theta = 1$.
Solving simple trig equations within a given range (e.g., $0^\circ \le \theta \le 360^\circ$ or $-\pi < x < \pi$).

1. The Three New Trigonometric Functions

In P2, we introduce the three reciprocal trigonometric functions. They are simply the multiplicative inverses (flips) of the functions you already know.

Definitions and Relationships

The new functions are Secant ($\sec \theta$), Cosecant ($\csc \theta$ or sometimes $\text{cosec } \theta$), and Cotangent ($\cot \theta$).

Secant: $\sec \theta = \frac{1}{\cos \theta}$
Cosecant: $\csc \theta = \frac{1}{\sin \theta}$
Cotangent: $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$

Memory Aid: The Co-Rule
The reciprocal function starts with 'co' if the original function doesn't, and vice versa.

$\mathbf{S}$ine is related to $\mathbf{C}$osecant.
$\mathbf{C}$osine is related to $\mathbf{S}$ecant.
$\mathbf{T}$angent is related to $\mathbf{C}$otangent.

Graphs and Asymptotes

When sketching these graphs, remember that whenever the original function equals zero, the reciprocal function is undefined, leading to an asymptote.

1. $y = \sec \theta$ (Reciprocal of $\cos \theta$)

Asymptotes: Occur whenever $\cos \theta = 0$, i.e., at $90^\circ, 270^\circ, 450^\circ, \dots$ or $\frac{\pi}{2}, \frac{3\pi}{2}, \dots$.
Range: Since $-1 \le \cos \theta \le 1$, the reciprocal function must satisfy $y \le -1$ or $y \ge 1$. The graph never exists between -1 and 1.

2. $y = \csc \theta$ (Reciprocal of $\sin \theta$)

Asymptotes: Occur whenever $\sin \theta = 0$, i.e., at $0^\circ, 180^\circ, 360^\circ, \dots$ or $0, \pi, 2\pi, \dots$.
Range: Similar to $\sec \theta$, the range is $y \le -1$ or $y \ge 1$.

Quick Review: Solving Reciprocal Equations

Always convert the reciprocal function back to $\sin$, $\cos$, or $\tan$.

Example: Solve $\sec x = 2$ for $0^\circ \le x \le 360^\circ$.

1. Rewrite: $\frac{1}{\cos x} = 2$

2. Rearrange: $\cos x = \frac{1}{2}$

3. Solve: $x = 60^\circ$ and $x = 360^\circ - 60^\circ = 300^\circ$.

Key Takeaway: The reciprocal functions allow us to deal with division in trigonometry. Their graphs show that they jump over the region between -1 and 1.

2. Pythagorean Identities (The New Generation)

We start with the foundational P1 identity: $\sin^2 \theta + \cos^2 \theta = 1$.

By dividing this identity by $\cos^2 \theta$ and then by $\sin^2 \theta$, we generate two new identities essential for P2 and beyond.

Identity 1: The Secant and Tangent Relationship

Divide $\sin^2 \theta + \cos^2 \theta = 1$ by $\mathbf{\cos^2 \theta}$:
$$ \frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} $$ $$ \tan^2 \theta + 1 = \sec^2 \theta $$

Formula 1: $\sec^2 \theta = 1 + \tan^2 \theta$

Identity 2: The Cosecant and Cotangent Relationship

Divide $\sin^2 \theta + \cos^2 \theta = 1$ by $\mathbf{\sin^2 \theta}$:
$$ \frac{\sin^2 \theta}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta} $$ $$ 1 + \cot^2 \theta = \csc^2 \theta $$

Formula 2: $\csc^2 \theta = 1 + \cot^2 \theta$

These two identities are primarily used for simplifying expressions or solving equations where $\tan$ and $\sec$, or $\cot$ and $\csc$, appear together.

Key Takeaway: The reciprocal identities allow you to swap between different trig functions, often making complicated expressions manageable. They are the keys to most P2 trigonometric proofs and simplification problems.

3. Compound Angle (Addition) Formulae

What happens if you need the sine of a sum of two angles, like $\sin(A+B)$? It is NOT $\sin A + \sin B$! We need the Compound Angle Formulae.

The Formulae (Memorise these!)

Sine Addition/Subtraction:

$$ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B $$

Cosine Addition/Subtraction:

$$ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B $$

Tangent Addition/Subtraction:

$$ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} $$

Memory Aid: Sine is friendly; Cosine is grumpy

Sine Formula ($\sin(A \pm B)$): Sine is "friendly"—it keeps the same sign ($\pm$) as the angle sum, and it mixes the functions ($\sin$ and $\cos$ together).
Cosine Formula ($\cos(A \pm B)$): Cosine is "grumpy"—it changes the sign ($\mp$), and it groups the functions (all $\cos$ together, all $\sin$ together).

Applications of Compound Angles

The primary uses are:

Exact Evaluation: Finding exact values for angles that aren't 30, 45, or 60 degrees. (e.g., $\cos 75^\circ = \cos(45^\circ + 30^\circ)$).
Simplification: Rewriting expressions like $\cos(x - 30^\circ)$ to make differentiation or integration easier (or simply to match required proof formats).

Did you know? The compound angle formulae are the foundation of almost all higher trigonometry. The double angle and R-formulae are simply special cases of these!

Key Takeaway: Compound angles allow you to separate a function of a sum into a mixture of functions of individual angles. Be extremely careful with the signs in the cosine formula!

4. Double Angle Formulae

These formulae are derived by setting $A = B$ in the compound angle formulae. They are incredibly useful for solving equations, simplifying expressions, and especially for integration (in P3/P2 Extension).

The Formulae

1. Sine Double Angle: (Setting $A=B$ in $\sin(A+B)$) $$ \sin 2A = 2 \sin A \cos A $$

2. Cosine Double Angle: (Setting $A=B$ in $\cos(A+B)$) $$ \cos 2A = \cos^2 A - \sin^2 A $$

Crucial Note on $\cos 2A$: Using the identity $\sin^2 A + \cos^2 A = 1$, we can substitute and get two other forms:

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

You must be familiar with all three forms! The forms involving $2 \cos^2 A - 1$ and $1 - 2 \sin^2 A$ are vital when dealing with powers of sine or cosine in calculus.

3. Tangent Double Angle: (Setting $A=B$ in $\tan(A+B)$) $$ \tan 2A = \frac{2 \tan A}{1 - \tan^2 A} $$

Common Mistake to Avoid

When solving an equation like $\sin 2x = \cos x$, do not divide both sides by $\cos x$ without considering the case where $\cos x = 0$!

Correct Method: Substitute $\sin 2x = 2 \sin x \cos x$:
$$ 2 \sin x \cos x = \cos x $$ $$ 2 \sin x \cos x - \cos x = 0 $$ $$ \cos x (2 \sin x - 1) = 0 $$ You now have two sets of solutions: $\cos x = 0$ and $\sin x = 1/2$. Never divide by a function that could be zero!

Key Takeaway: Double angle formulae relate a function of $2A$ to functions of $A$. Remember the three versions of $\cos 2A$.

5. The R-Formula (Auxiliary Angle Form)

The R-formula is one of the most distinctive and important skills in P2 Trigonometry. It allows us to rewrite a linear combination of sine and cosine ($a \sin \theta + b \cos \theta$) into a single, combined wave function.

Goal: Express $a \sin \theta + b \cos \theta$ in the form $R \sin(\theta \pm \alpha)$ or $R \cos(\theta \pm \alpha)$.

Analogy: Imagine two waves hitting a wall. The R-formula helps you calculate the one single resulting wave (its peak height, $R$, and its phase shift, $\alpha$).

Steps to Find R and $\alpha$

Let’s use the target form $R \sin(\theta + \alpha)$ where $R > 0$ and $0^\circ < \alpha < 90^\circ$ (or $0 < \alpha < \pi/2$).

Step 1: Expand the target form.

Using the Compound Angle Formula:

$$ R \sin(\theta + \alpha) = R(\sin \theta \cos \alpha + \cos \theta \sin \alpha) $$ $$ R \sin(\theta + \alpha) = (R \cos \alpha) \sin \theta + (R \sin \alpha) \cos \theta $$

Step 2: Compare coefficients.

We compare this expanded form to our original expression: $a \sin \theta + b \cos \theta$.

Coefficient of $\sin \theta$: $R \cos \alpha = a$ (Equation I)
Coefficient of $\cos \theta$: $R \sin \alpha = b$ (Equation II)

Step 3: Calculate R (The Amplitude).

Square and add Equations I and II, then use $\cos^2 \alpha + \sin^2 \alpha = 1$:

$$ (R \cos \alpha)^2 + (R \sin \alpha)^2 = a^2 + b^2 $$ $$ R^2 (\cos^2 \alpha + \sin^2 \alpha) = a^2 + b^2 $$ $$ R^2 (1) = a^2 + b^2 $$

Result: $R = \sqrt{a^2 + b^2}$

Step 4: Calculate $\alpha$ (The Phase Shift).

Divide Equation II by Equation I:

$$ \frac{R \sin \alpha}{R \cos \alpha} = \frac{b}{a} $$

Result: $\tan \alpha = \frac{b}{a}$

Crucial Point: When calculating $\alpha$, you must use $a$ and $b$ from the original expression $a \sin \theta + b \cos \theta$ to determine the signs of $R \sin \alpha$ and $R \cos \alpha$. Since $R$ is always positive, the signs of $a$ and $b$ determine the quadrant of $\alpha$.

Example: Express $4 \cos \theta - 3 \sin \theta$ in the form $R \cos(\theta + \alpha)$.

1. Expand $R \cos(\theta + \alpha) = R(\cos \theta \cos \alpha - \sin \theta \sin \alpha)$
2. Compare to $4 \cos \theta - 3 \sin \theta$:

$R \cos \alpha = 4$ (positive)
$R \sin \alpha = 3$ (positive - we ignore the negative sign from the original expression because it was absorbed by the expansion sign change).

3. $R = \sqrt{4^2 + (-3)^2} = \sqrt{25} = 5$
4. $\tan \alpha = \frac{3}{4}$. (Since $R \sin \alpha$ and $R \cos \alpha$ are both positive, $\alpha$ is in the first quadrant.)
$\alpha = \tan^{-1}(3/4) \approx 36.9^\circ$
Answer: $5 \cos(\theta + 36.9^\circ)$

Why do we use the R-Formula?

We use it to solve complex trigonometric equations or find maximum/minimum values.

If $f(\theta) = R \sin(\theta + \alpha)$, then the maximum value is $R$, and the minimum value is $-R$.
The equation $a \sin \theta + b \cos \theta = k$ becomes $R \sin(\theta \pm \alpha) = k$, which is a simple P1-style equation to solve.

Key Takeaway: The R-formula converts a sum of sine and cosine into a single, manageable wave. $R$ is found using Pythagoras, and $\alpha$ is found using tangent, making sure to verify the quadrant of $\alpha$.

Final Summary Checklist for P2 Trigonometry

You must be fluent in applying the following:

Reciprocals: $\sec, \csc, \cot$ and their graphs/asymptotes.
Pythagorean Identities: $\sec^2 \theta = 1 + \tan^2 \theta$ and $\csc^2 \theta = 1 + \cot^2 \theta$.
Compound Angles: $\sin(A \pm B)$ and $\cos(A \pm B)$ (Remember Sine is friendly, Cosine is grumpy!).
Double Angles: $\sin 2A$ and all three forms of $\cos 2A$.
R-Formula: The systematic process for converting $a \sin \theta + b \cos \theta$ into the auxiliary angle form $R \sin(\theta \pm \alpha)$ or $R \cos(\theta \pm \alpha)$.

Practice converting these identities backward and forward—that’s where the flexibility in solving P2 problems comes from!

References

Cambridge International AS & A Level Mathematics (9709) Syllabus for examination in 2026 and 2027 — Cambridge University Press & Assessment
Cambridge International AS & A Level Mathematics 9709 Scheme of Work: Pure Mathematics 2 — Cambridge Assessment International Education

Quick check

Can you answer these now?

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

What are the definitions of the reciprocal trigonometric functions secant, cosecant, and cotangent?

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

What is the "Co-Rule" memory aid for reciprocal functions?

The reciprocal function starts with "co" if the original does not, and vice versa: Sine $\leftrightarrow$ Cosecant, Cosine $\leftrightarrow$ Secant, and Tangent $\leftrightarrow$ Cotangent.

Where do the asymptotes occur on the graphs of reciprocal trigonometric functions?

Asymptotes occur wherever the original function equals zero, as division by zero is undefined (e.g., $\sec \theta$ has asymptotes where $\cos \theta = 0$).

What is the general strategy for solving equations involving reciprocal trigonometric functions?

Rewrite the reciprocal function as its primary counterpart (sine, cosine, or tangent) and solve the resulting equation.

What is the Pythagorean identity relating secant and tangent?

$\sec^2 \theta = 1 + \tan^2 \theta$

What is the Pythagorean identity relating cosecant and cotangent?

$\csc^2 \theta = 1 + \cot^2 \theta$

What are the compound angle formulae for sine and cosine?

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ and $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$.

What is the compound angle formula for tangent?

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

What is the double angle formula for sine?

$\sin 2A = 2 \sin A \cos A$

What are the three different forms of the cosine double angle formula?

1) $\cos 2A = \cos^2 A - \sin^2 A$
2) $\cos 2A = 2 \cos^2 A - 1$
3) $\cos 2A = 1 - 2 \sin^2 A$

What is the double angle formula for tangent?

$\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}$

In the R-formula, how are $R$ and $\alpha$ calculated for the form $R \sin(\theta + \alpha)$?

$R = \sqrt{a^2 + b^2}$ and $\tan \alpha = \frac{b}{a}$.

How is the R-formula used to find the maximum or minimum value of an expression?

For an expression rewritten as $R \sin(\theta \pm \alpha)$, the maximum value is $R$ and the minimum value is $-R$.

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