Trigonometry

Q: What are the three reciprocal trigonometric identities?

The three reciprocal identities are $ \sec x \equiv \frac{1}{\cos x} $, $ \csc x \equiv \frac{1}{\sin x} $, and $ \cot x \equiv \frac{1}{\tan x} \equiv \frac{\cos x}{\sin x} $.

Q: What are the Pythagorean identities involving secant and cosecant?

$ 1 + \tan^2 x \equiv \sec^2 x $ and $ 1 + \cot^2 x \equiv \csc^2 x $.

Q: What are the expansion formulae for the sine of a sum or difference of two angles?

$ \sin(A + B) \equiv \sin A \cos B + \cos A \sin B $ and $ \sin(A - B) \equiv \sin A \cos B - \cos A \sin B $.

Q: What are the expansion formulae for the cosine of a sum or difference of two angles?

$ \cos(A + B) \equiv \cos A \cos B - \sin A \sin B $ and $ \cos(A - B) \equiv \cos A \cos B + \sin A \sin B $.

Q: What is the formula for expanding the tangent of a sum or difference of two angles?

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

Q: What is the double angle identity for $ \sin 2A $?

$ \sin 2A \equiv 2 \sin A \cos A $.

Q: What are the three forms of the $ \cos 2A $ double angle identity?

1. $ \cos 2A \equiv \cos^2 A - \sin^2 A $ 2. $ \cos 2A \equiv 2 \cos^2 A - 1 $ 3. $ \cos 2A \equiv 1 - 2 \sin^2 A $

Q: What is the double angle identity for $ \tan 2A $?

$ \tan 2A \equiv \frac{2 \tan A}{1 - \tan^2 A} $.

Q: What is the general strategy for solving complex trigonometric equations in P3?

Manipulate the equation using identities until it involves only one single angle and one single trigonometric function .

Q: Why should you factorize instead of dividing by a trigonometric function when solving equations?

Dividing both sides by a function (like $\sin x$) will cause you to lose all solutions where that function equals zero.

Practice

Mathematics (YMA01) · Pearson Edexcel International A Level · 9 min read · Updated: 7 Sept 2026

Welcome to Pure Mathematics 3: Advanced Trigonometry!

Hello future mathematician! You’ve already mastered the basics of trigonometry in P1 and P2—things like SOH CAH TOA, the sine and cosine rules, and the fundamental identities ($\tan\theta \equiv \frac{\sin\theta}{\cos\theta}$ and $\sin^2\theta + \cos^2\theta \equiv 1$).

In P3, we move beyond those basics. We are going to learn powerful new tools—called identities—that allow us to simplify incredibly complicated expressions, solve equations you previously thought impossible, and analyze wave behavior much more effectively.

Don't worry if this seems tricky at first; we will break down every identity step-by-step. Let's dive into the world of trig manipulation!

Quick Review: Prerequisite Identities

Before we start, let’s refresh the basics we will use constantly:

Reciprocal Identities:
$ \sec x \equiv \frac{1}{\cos x} $
$ \csc x \equiv \frac{1}{\sin x} $
$ \cot x \equiv \frac{1}{\tan x} \equiv \frac{\cos x}{\sin x} $
Pythagorean Identities (Derived from $\sin^2\theta + \cos^2\theta \equiv 1$):
$ 1 + \tan^2 x \equiv \sec^2 x $
$ 1 + \cot^2 x \equiv \csc^2 x $

Key Takeaway: When simplifying expressions, always look to substitute these identities to reduce the number of terms or functions.

Section 1: The Addition (or Compound Angle) Formulae

Have you ever wondered what $\sin(A+B)$ is? It’s definitely not $\sin A + \sin B$! The addition formulae tell us exactly how to expand trigonometric functions when the input is a sum or difference of two angles. These four identities are the foundation of P3 Trigonometry.

The Four Addition Formulae

Remember, these are given in your formula booklet, but understanding how they work is vital.

Sine Addition:
$ \sin(A + B) \equiv \sin A \cos B + \cos A \sin B $
Sine Subtraction:
$ \sin(A - B) \equiv \sin A \cos B - \cos A \sin B $
Cosine Addition:
$ \cos(A + B) \equiv \cos A \cos B - \sin A \sin B $
Cosine Subtraction:
$ \cos(A - B) \equiv \cos A \cos B + \sin A \sin B $

💡 Memory Aid for Sine and Cosine

This trick helps you remember the structure:

Sine (Sin): Sine is "friendly." It mixes functions (Sin Cos, Cos Sin) and keeps the sign (+ stays +, - stays -).
Cosine (Cos): Cosine is "selfish." It groups functions (Cos Cos, Sin Sin) and changes the sign (+ becomes -, - becomes +).

Tangent Addition Formula

This identity is derived by dividing $\sin(A+B)$ by $\cos(A+B)$:

$ \tan(A + B) \equiv \frac{\tan A + \tan B}{1 - \tan A \tan B} $

$ \tan(A - B) \equiv \frac{\tan A - \tan B}{1 + \tan A \tan B} $

Example: Use the addition formula to find the exact value of $\cos(105^\circ)$.
We know that $105^\circ = 60^\circ + 45^\circ$. We use the Cosine Addition formula:
$ \cos(105^\circ) = \cos(60^\circ + 45^\circ) $
$ = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ $
$ = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $
$ = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = \frac{\sqrt{2} - \sqrt{6}}{4} $

Key Takeaway: Addition formulae allow us to break down complex angles into combinations of simple, known angles (like 30°, 45°, 60°).

Section 2: The Double Angle Formulae

The double angle formulae are perhaps the most frequently used identities in P3. They are simply a special case of the addition formulae where $A = B$.

Don't worry, you don't need to memorize them all, but you must know how to choose the right one for the job!

1. The Sine Double Angle Formula

If we take $\sin(A+B)$ and set $B=A$:

$ \sin(A + A) = \sin A \cos A + \cos A \sin A $

Identity: $ \sin 2A \equiv 2 \sin A \cos A $

2. The Cosine Double Angle Formulae (Three Forms!)

If we take $\cos(A+B)$ and set $B=A$:

$ \cos(A + A) = \cos A \cos A - \sin A \sin A $

Form 1: $ \cos 2A \equiv \cos^2 A - \sin^2 A $

Why three forms? Because we can substitute the Pythagorean identity $\sin^2 A + \cos^2 A \equiv 1$ to eliminate either the $\sin^2$ or the $\cos^2$ term.

Form 2 (In terms of Cosine only):
Substitute $\sin^2 A = 1 - \cos^2 A$ into Form 1:
$ \cos 2A \equiv \cos^2 A - (1 - \cos^2 A) $
Identity: $ \cos 2A \equiv 2 \cos^2 A - 1 $

Form 3 (In terms of Sine only):
Substitute $\cos^2 A = 1 - \sin^2 A$ into Form 1:
$ \cos 2A \equiv (1 - \sin^2 A) - \sin^2 A $
Identity: $ \cos 2A \equiv 1 - 2 \sin^2 A $

When to use which form?

Use Form 2 ($2 \cos^2 A - 1$) if your equation already has lots of $\cos$ terms.
Use Form 3 ($1 - 2 \sin^2 A$) if your equation already has lots of $\sin$ terms.
Use Form 1 ($\cos^2 A - \sin^2 A$) usually when factorizing or proving other identities.

3. The Tangent Double Angle Formula

Setting $A=B$ in the Tangent Addition Formula gives:

Identity: $ \tan 2A \equiv \frac{2 \tan A}{1 - \tan^2 A} $

Key Takeaway: The double angle identities allow us to rewrite terms involving $2x$ in terms of functions of $x$, which is essential for solving equations.

Section 3: Solving Trigonometric Equations (The P3 Way)

In P3, equations will involve a mix of angles ($x$ and $2x$) and functions ($\sin, \cos, \tan$). Your goal is always the same: get the equation into one single angle and one single function.

Step-by-Step: Solving Equations with $2x$

Example: Solve $2 \cos 2\theta - \cos \theta + 1 = 0$ for $0^\circ \le \theta < 360^\circ$.

Identify the Problem: We have $\cos 2\theta$ (angle $2\theta$) and $\cos \theta$ (angle $\theta$). We must convert $\cos 2\theta$ to angle $\theta$.
Choose the Right Identity: Since the rest of the equation uses $\cos \theta$, we choose the identity for $\cos 2\theta$ that only involves cosine (Form 2): $2 \cos^2 \theta - 1$.
Substitute and Simplify:
$ 2(2 \cos^2 \theta - 1) - \cos \theta + 1 = 0 $
$ 4 \cos^2 \theta - 2 - \cos \theta + 1 = 0 $
$ 4 \cos^2 \theta - \cos \theta - 1 = 0 $
Solve the Quadratic: This is now a quadratic equation in terms of $C = \cos \theta$. Since it doesn't factorize easily, use the quadratic formula:
$ \cos \theta = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(-1)}}{2(4)} = \frac{1 \pm \sqrt{17}}{8} $
Find the Solutions: Calculate the two values and find the principal angles, then find all solutions in the given range (using CAST or graph method).

⚠️ Common Mistake: Losing Solutions

If you encounter an equation like $\sin 2x = \sin x$, DO NOT divide both sides by $\sin x$. If you divide, you lose the solutions where $\sin x = 0$!

Correct Method:

Use the double angle identity: $2 \sin x \cos x = \sin x$
Rearrange and Factorize: $2 \sin x \cos x - \sin x = 0$
Factor out $\sin x$: $\sin x (2 \cos x - 1) = 0$
Solve each factor: $\sin x = 0$ AND $2 \cos x - 1 = 0$.

Key Takeaway: Always aim for factorisation when solving equations, not division, to ensure you find all roots.

Section 4: The R-Formula (Combining Trig Functions)

This is one of the biggest and most important applications of the P3 identities. The $R$-formula allows us to transform an expression involving a combination of sine and cosine terms (like $a\cos\theta + b\sin\theta$) into a single, simpler sine or cosine function (like $R\cos(\theta \pm \alpha)$ or $R\sin(\theta \pm \alpha)$).

Analogy: The Wave Simplifier

Imagine you have two musical notes (a sine wave and a cosine wave) playing simultaneously. The resulting sound wave is complex. The $R$-formula combines those two complex inputs into a single, clean wave defined by its Amplitude ($R$) and its Phase Shift ($\alpha$).

The Transformation Process

We want to express $a\cos\theta + b\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R>0$ and $0^\circ < \alpha < 90^\circ$ (or the required range).

Let’s transform $3\cos\theta + 4\sin\theta$ into the form $R\cos(\theta - \alpha)$.

Choose the Target Identity: We expand the target form using the addition formula (Cosine Subtraction):
$ R\cos(\theta - \alpha) \equiv R(\cos\theta \cos\alpha + \sin\theta \sin\alpha) $
$ R\cos(\theta - \alpha) \equiv (R\cos\alpha)\cos\theta + (R\sin\alpha)\sin\theta $
Compare Coefficients: Now, compare this expanded form to the original expression $3\cos\theta + 4\sin\theta$:
Coefficient of $\cos\theta$: $R\cos\alpha = 3$ (Equation 1)
Coefficient of $\sin\theta$: $R\sin\alpha = 4$ (Equation 2)
Find R (Amplitude): Square both equations and add them together. This uses the Pythagorean identity!
$(R\cos\alpha)^2 + (R\sin\alpha)^2 = 3^2 + 4^2$
$R^2(\cos^2\alpha + \sin^2\alpha) = 9 + 16$
$R^2(1) = 25 \implies R = 5$
Find $\alpha$ (Phase Shift): Divide Equation 2 by Equation 1:
$ \frac{R\sin\alpha}{R\cos\alpha} = \frac{4}{3} $
$ \tan\alpha = \frac{4}{3} $
$ \alpha = \arctan\left(\frac{4}{3}\right) \approx 53.13^\circ $

Conclusion: $3\cos\theta + 4\sin\theta \equiv 5\cos(\theta - 53.13^\circ)$.

Important Note on R

$R$ is always found using Pythagoras' Theorem:
$ R = \sqrt{a^2 + b^2} $

$R$ is the maximum value (amplitude) that the expression can achieve, and $-R$ is the minimum value.

Solving Maximum/Minimum Problems using R-Formula

Once you have the expression in the $R$-form (e.g., $5\cos(\theta - 53.13^\circ)$), solving max/min questions becomes easy.

Maximum Value: The maximum value of any cosine function is $+1$.
Maximum expression value is $5 \times (+1) = 5$.
This happens when the inside angle is $0^\circ, 360^\circ, \ldots$: $\theta - 53.13^\circ = 0 \implies \theta = 53.13^\circ$.
Minimum Value: The minimum value of any cosine function is $-1$.
Minimum expression value is $5 \times (-1) = -5$.
This happens when the inside angle is $180^\circ, 540^\circ, \ldots$: $\theta - 53.13^\circ = 180^\circ \implies \theta = 233.13^\circ$.

Key Takeaway: The $R$-formula converts complex sums into a simple form, making it easy to find maximum/minimum values and solve equations like $a\cos\theta + b\sin\theta = k$.

Summary and Next Steps

Congratulations! You've covered the heart of P3 Trigonometry. Remember, mastery here comes from practice. Every time you see a $\cos 2x$ or a messy sum like $3\sin x - \cos x$, you should immediately reach for your P3 toolbox!

Quick P3 Trigonometry Checklist

Can I use the Addition Formulae to find the exact value of $\sin 15^\circ$?
Can I rewrite $\frac{\sin 2x}{1 + \cos 2x}$ using the Double Angle Formulae and simplify it to $\tan x$?
Can I choose the correct form of $\cos 2x$ to simplify a quadratic trigonometric equation?
Can I express $a\sin\theta - b\cos\theta$ in the form $R\sin(\theta - \alpha)$ and determine its maximum value?

Keep practising these transformations, and you’ll master this section!

Quick check

Can you answer these now?

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

What are the three reciprocal trigonometric identities?

The three reciprocal identities are $ \sec x \equiv \frac{1}{\cos x} $, $ \csc x \equiv \frac{1}{\sin x} $, and $ \cot x \equiv \frac{1}{\tan x} \equiv \frac{\cos x}{\sin x} $.

What are the Pythagorean identities involving secant and cosecant?

$ 1 + \tan^2 x \equiv \sec^2 x $ and $ 1 + \cot^2 x \equiv \csc^2 x $.

What are the expansion formulae for the sine of a sum or difference of two angles?

$ \sin(A + B) \equiv \sin A \cos B + \cos A \sin B $ and $ \sin(A - B) \equiv \sin A \cos B - \cos A \sin B $.

What are the expansion formulae for the cosine of a sum or difference of two angles?

$ \cos(A + B) \equiv \cos A \cos B - \sin A \sin B $ and $ \cos(A - B) \equiv \cos A \cos B + \sin A \sin B $.

What is the formula for expanding the tangent of a sum or difference of two angles?

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

What is the double angle identity for $ \sin 2A $?

$ \sin 2A \equiv 2 \sin A \cos A $.

What are the three forms of the $ \cos 2A $ double angle identity?

1. $ \cos 2A \equiv \cos^2 A - \sin^2 A $
2. $ \cos 2A \equiv 2 \cos^2 A - 1 $
3. $ \cos 2A \equiv 1 - 2 \sin^2 A $

What is the double angle identity for $ \tan 2A $?

$ \tan 2A \equiv \frac{2 \tan A}{1 - \tan^2 A} $.

What is the general strategy for solving complex trigonometric equations in P3?

Manipulate the equation using identities until it involves only one single angle and one single trigonometric function.

Why should you factorize instead of dividing by a trigonometric function when solving equations?

Dividing both sides by a function (like $\sin x$) will cause you to lose all solutions where that function equals zero.

What is the purpose of using the $R$-formula?

To transform a combination of sine and cosine waves, $ a\cos\theta + b\sin\theta $, into a single sine or cosine wave.

How do you calculate the values of $R$ and $\alpha$ in the transformation $ a\cos\theta + b\sin\theta \equiv R\cos(\theta - \alpha) $?

$ R = \sqrt{a^2 + b^2} $ and $ \alpha = \arctan\left(\frac{b}{a}\right) $ (by comparing coefficients).

How do you find the maximum and minimum values of an expression using the $R$-formula?

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

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