🧠 Further Pure Mathematics Study Notes: Advanced Trigonometry

Hello future Mathematicians! Welcome to the exciting world of Further Pure Trigonometry. You already know the basics (SOH CAH TOA and the CAST diagram), but in this chapter, we level up!

We are going to learn powerful formulas that allow us to combine, split, and transform trigonometric functions. These tools are absolutely vital for solving complex equations, proving tricky identities, and modelling real-world phenomena like sound waves and electrical circuits. Don't worry if some concepts look intimidating at first—we'll break them down step-by-step! Let's get started!

Review Box: Core Identities (The Basics You MUST Know)

Before diving into the "Further" content, let's quickly remind ourselves of the essentials:

  • Pythagorean Identity: \(\sin^2\theta + \cos^2\theta = 1\)
  • Derived Pythagorean Identities:
    (1) \(1 + \tan^2\theta = \sec^2\theta\)
    (2) \(\cot^2\theta + 1 = \csc^2\theta\)
  • Tangent Identity: \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)

Section 1: The Compound Angle Formulae (Addition Formulae)

The Compound Angle Formulae allow us to find the trigonometric ratio of an angle that is the sum or difference of two known angles (e.g., finding \(\sin(75^\circ)\) by using \(\sin(45^\circ + 30^\circ)\)).

This is the foundation of almost everything else in this chapter!

1.1 The Formulae You Need to Memorise

Sine Addition/Subtraction
\(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
\(\sin(A - B) = \sin A \cos B - \cos A \sin B\)

Cosine Addition/Subtraction
\(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
\(\cos(A - B) = \cos A \cos B + \sin A \sin B\)

Tangent Addition/Subtraction
\(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
\(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\)

🧠 Memory Aid: Sine vs. Cosine Personality

Think of the functions as having personalities:

  • Sine (\(\sin\)): The Social function. It shares! It alternates partners (sin, cos, cos, sin) and keeps the same sign (\(+ \rightarrow +\), \(- \rightarrow -\)).
  • Cosine (\(\cos\)): The Selfish function. It pairs up with itself first (cos, cos, sin, sin) and changes the sign (\(+ \rightarrow -\), \(- \rightarrow +\)).

1.2 Application Example

Example: Simplify the expression \(\cos(x + 90^\circ)\).

We use the \(\cos(A + B)\) formula, where \(A=x\) and \(B=90^\circ\):
\(\cos(x + 90^\circ) = \cos x \cos 90^\circ - \sin x \sin 90^\circ\)
We know that \(\cos 90^\circ = 0\) and \(\sin 90^\circ = 1\).
\(\cos(x + 90^\circ) = (\cos x)(0) - (\sin x)(1)\)
\(\cos(x + 90^\circ) = -\sin x\)

Key Takeaway (Compound Angles): These formulae allow us to expand or combine expressions involving sums/differences of angles. Pay crucial attention to the sign change in the cosine formulas.

Section 2: Doubling the Fun – Double Angle Formulae

If we set \(A = B = \theta\) in the Compound Angle Formulae, we instantly get the Double Angle Formulae. These are essential for simplifying expressions and solving equations involving \(\sin 2\theta\) or \(\cos 2\theta\).

2.1 Sine Double Angle

\(\sin 2\theta = 2 \sin \theta \cos \theta\)

Did you know? This formula is very powerful for replacing terms in equations where you have a mix of angles (\(2\theta\) and \(\theta\)).

2.2 Cosine Double Angle (The Triple Threat)

The Cosine Double Angle formula is unique because it has three equally valid forms. You must know all three, as choosing the right one can make solving an equation dramatically easier!

Form 1 (The Original): \(\cos 2\theta = \cos^2\theta - \sin^2\theta\)

Using the identity \(\sin^2\theta = 1 - \cos^2\theta\), we get:

Form 2 (Cosine only): \(\cos 2\theta = 2\cos^2\theta - 1\)

Using the identity \(\cos^2\theta = 1 - \sin^2\theta\), we get:

Form 3 (Sine only): \(\cos 2\theta = 1 - 2\sin^2\theta\)

2.3 Tangent Double Angle

\(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\)

🔥 Common Mistake Alert!

DO NOT confuse \(\sin 2\theta\) with \(2 \sin \theta\). They are NOT the same!
Example: If \(\theta = 30^\circ\):
\(\sin 2\theta = \sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866\)
\(2 \sin \theta = 2 \sin 30^\circ = 2(\frac{1}{2}) = 1\)

Key Takeaway (Double Angles): The primary use is Identity Proofs and converting expressions involving \(\cos 2\theta\) into terms involving only \(\cos\theta\) or only \(\sin\theta\).

Section 3: The Trigonometric Transformer – The R-Formula

The R-Formula (also known as the Auxiliary Angle method) is one of the most powerful techniques in Further Pure Maths. It allows you to take an expression that is a sum of two waves (e.g., \(3\cos\theta + 4\sin\theta\)) and combine it into a single, simplified wave expression (e.g., \(R\cos(\theta - \alpha)\)).

3.1 What is the R-Formula and Why Do We Use It?

We want to express:
$$a\cos\theta + b\sin\theta$$ in the form:
$$R\cos(\theta \pm \alpha) \text{ or } R\sin(\theta \pm \alpha)$$

The main goals of using the R-Formula are:

  • To find the maximum and minimum values of the expression.
  • To solve complex trigonometric equations that involve both \(\sin\theta\) and \(\cos\theta\).

🌊 Analogy: Combining Waves

Imagine two sound waves (cosine and sine) hitting your ear simultaneously. The R-Formula takes those two separate waves and mathematically represents them as a single, combined wave, defining its new amplitude (\(R\)) and its phase shift (\(\alpha\)).

3.2 Step-by-Step Process for Finding R and \(\alpha\)

Let's use the form \(R\cos(\theta - \alpha)\) for demonstration.
We start by setting the expression equal to its expanded form:
$$a\cos\theta + b\sin\theta = R\cos(\theta - \alpha)$$

Step 1: Expand the Right Side (RHS)
Use the Compound Angle formula for \(\cos(\theta - \alpha)\):
$$R\cos(\theta - \alpha) = R(\cos\theta \cos\alpha + \sin\theta \sin\alpha)$$
$$R\cos(\theta - \alpha) = (R\cos\alpha)\cos\theta + (R\sin\alpha)\sin\theta$$

Step 2: Compare Coefficients
We compare the coefficients of \(\cos\theta\) and \(\sin\theta\) on both sides:

  • Coefficient of \(\cos\theta\): \(a = R\cos\alpha\) (Equation 1)
  • Coefficient of \(\sin\theta\): \(b = R\sin\alpha\) (Equation 2)

Step 3: Find R (The Amplitude)
Square Equations 1 and 2, and add them together:
$$a^2 + b^2 = (R\cos\alpha)^2 + (R\sin\alpha)^2$$ $$a^2 + b^2 = R^2(\cos^2\alpha + \sin^2\alpha)$$
Since \(\cos^2\alpha + \sin^2\alpha = 1\):

The Amplitude R: \(R = \sqrt{a^2 + b^2}\)
R is always positive.

Step 4: Find \(\alpha\) (The Phase Shift)
Divide Equation 2 by Equation 1:
$$\frac{R\sin\alpha}{R\cos\alpha} = \frac{b}{a}$$
$$\tan\alpha = \frac{b}{a}$$

The Angle \(\alpha\): \(\alpha = \arctan\left(\frac{b}{a}\right)\)

Remember to express \(\alpha\) in the required range (usually \(0^\circ < \alpha < 90^\circ\)).

3.3 Application: Maximum and Minimum Values

Once you have the form \(R\cos(\theta - \alpha)\), finding the max/min is easy:

  • The maximum value occurs when the cosine/sine part is 1. Max Value = \(R\).
  • The minimum value occurs when the cosine/sine part is -1. Min Value = \(-R\).
Key Takeaway (R-Formula): Use the R-Formula to simplify \(a\cos\theta + b\sin\theta\) into a single trigonometric function. Remember \(R\) is Pythagorean (\(R = \sqrt{a^2 + b^2}\)) and \(\alpha\) uses tangent (\(\tan\alpha = b/a\)).

Section 4: Solving Advanced Trigonometric Equations

In Further Pure Maths, solving equations often requires using the identities we just learned before we can solve them using standard methods (like the CAST diagram).

4.1 Strategy 1: Using Double Angles to Simplify

If an equation mixes \(\cos 2\theta\) and \(\cos\theta\) (or \(\sin\theta\)), you must convert the double angle term.

Example: Solve \(2\cos 2\theta - \sin\theta = 1\) for \(0^\circ \le \theta \le 360^\circ\).

Step 1: Choose the correct identity.
Since the equation contains \(\sin\theta\), we should replace \(\cos 2\theta\) with the sine-only form (\(1 - 2\sin^2\theta\)).

Step 2: Substitute and Rearrange.
$$2(1 - 2\sin^2\theta) - \sin\theta = 1$$ $$2 - 4\sin^2\theta - \sin\theta = 1$$
Rearrange into a quadratic equation by moving all terms to one side: $$0 = 4\sin^2\theta + \sin\theta - 1$$

Step 3: Solve the Quadratic.
Let \(x = \sin\theta\). Solve \(4x^2 + x - 1 = 0\) (often requires the quadratic formula).

Step 4: Find the angles.
Once you find the values for \(\sin\theta\), use the inverse sine function and the CAST diagram to find all solutions in the given range.

4.2 Strategy 2: Using the R-Formula to Solve Equations

If an equation is in the form \(a\cos\theta + b\sin\theta = c\), use the R-Formula.

Example: Solve \(3\cos\theta + 4\sin\theta = 2\) for \(0^\circ \le \theta \le 360^\circ\).

Step 1: Convert to R-Form.
Using the steps from Section 3, let \(3\cos\theta + 4\sin\theta = R\cos(\theta - \alpha)\).
\(R = \sqrt{3^2 + 4^2} = 5\).
\(\tan\alpha = \frac{4}{3} \implies \alpha \approx 53.13^\circ\).
So, \(5\cos(\theta - 53.13^\circ) = 2\).

Step 2: Isolate the Single Function.
$$\cos(\theta - 53.13^\circ) = \frac{2}{5} = 0.4$$

Step 3: Solve for the Combined Angle.
Let \(\phi = \theta - 53.13^\circ\). We solve \(\cos\phi = 0.4\).
Base angle: \(\cos^{-1}(0.4) \approx 66.42^\circ\).
Since cosine is positive, solutions are in Quadrant I and IV.
\(\phi_1 = 66.42^\circ\)
\(\phi_2 = 360^\circ - 66.42^\circ = 293.58^\circ\)
(Important: Always check if you need to find negative solutions or solutions beyond 360° based on your new range!)

Step 4: Find \(\theta\).
$$\theta = \phi + 53.13^\circ$$
\(\theta_1 = 66.42^\circ + 53.13^\circ = 119.55^\circ\)
\(\theta_2 = 293.58^\circ + 53.13^\circ = 346.71^\circ\)

🎯 Accessibility Tip: Range Transformation

When solving equations like \(\sin(2\theta)\) or \(\cos(\theta - \alpha)\), remember to transform the original range before solving for the composite angle!
If \(0^\circ \le \theta \le 360^\circ\), then \(0^\circ \le 2\theta \le 720^\circ\).

Key Takeaway (Solving Equations): The key is simplification. If you see mixed terms, use the identities (Compound or Double Angle) to turn the equation into a single trigonometric function or a standard quadratic form.

🎉 Conclusion and Next Steps

Congratulations! You've navigated the three pillars of Further Pure Trigonometry: Compound Angles, Double Angles, and the R-Formula. These topics are challenging, but mastering them makes you incredibly powerful in solving higher-level mathematical problems. Practice, practice, practice! Work through examples of identity proofs and complex equation solving—that is where the real understanding sticks! You've got this!