Mathematics (9709) | Trigonometry

Trigonometry (Pure Mathematics 3: 9709) Study Notes

Welcome to the Pure Mathematics 3 chapter on Trigonometry! If you found the trig in P1 straightforward, get ready to expand your toolkit. P3 Trigonometry builds heavily on your basic identities and adds powerful new functions and formulae, notably the R-formula, which is essential for combining waves. Mastering this topic is key for Paper 3 success!

1. The Reciprocal Functions: Secant, Cosecant, and Cotangent

In P3, we introduce three new trigonometric functions. These are simply the reciprocals of the familiar ones.

1.1 Definitions and Relationships

• Secant (\(\sec \theta\)): The reciprocal of cosine.
  $$ \sec \theta = \frac{1}{\cos \theta} $$
• Cosecant (\(\csc \theta\) or \(\mathrm{cosec} \theta\)): The reciprocal of sine.
  $$ \csc \theta = \frac{1}{\sin \theta} $$
• Cotangent (\(\cot \theta\)): The reciprocal of tangent.
  $$ \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} $$

Memory Aid: Notice the helpful confusion avoidance: the function starting with 'C' is the reciprocal of the function starting with 'S', and vice versa. (Cosecant goes with Sine; Secant goes with Cosine).

1.2 Graphs, Domains, and Ranges

Since these functions involve division by $\sin \theta$ or $\cos \theta$, their graphs will feature vertical asymptotes wherever the original function is zero.

• \(\mathbf{y = \sec \theta}\): Asymptotes occur where \(\cos \theta = 0\). (e.g., \(\pm 90^\circ\), \(\pm 270^\circ\)). Range: \(\sec \theta \leq -1\) or \(\sec \theta \geq 1\).
• \(\mathbf{y = \csc \theta}\): Asymptotes occur where \(\sin \theta = 0\). (e.g., \(0^\circ\), \(\pm 180^\circ\), \(\pm 360^\circ\)). Range: \(\csc \theta \leq -1\) or \(\csc \theta \geq 1\).
• \(\mathbf{y = \cot \theta}\): Asymptotes occur where \(\sin \theta = 0\). Period is \(\pi\) (or \(180^\circ\)). Range: All real numbers.

Key Takeaway: Sketching the graph of the underlying function (\(\sin \theta\) or \(\cos \theta\)) first is the best way to determine the location of the asymptotes and the shape of the reciprocal graph.

2. Advanced Pythagorean Identities

We build upon the fundamental identity: \(\sin^2 \theta + \cos^2 \theta = 1\).

2.1 Deriving the New Identities

1. Divide by \(\cos^2 \theta\):

$$ \frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} $$

This gives us the first P3 identity:

$$ \mathbf{\tan^2 \theta + 1 = \sec^2 \theta} $$

2. Divide by \(\sin^2 \theta\):

$$ \frac{\sin^2 \theta}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta} $$

This gives us the second P3 identity:

$$ \mathbf{1 + \cot^2 \theta = \csc^2 \theta} $$

2.2 Using the Identities in Simplification and Proof

These identities are vital for simplifying complex expressions and solving equations where reciprocal functions are involved. Always try to convert everything back to \(\sin \theta\) and \(\cos \theta\) if simplification seems difficult.

Example: To solve \(\sec^2 \theta = 3 \tan \theta - 1\), you must first replace \(\sec^2 \theta\) with \((1 + \tan^2 \theta)\) to create a solvable quadratic in \(\tan \theta\).

Key Takeaway: When solving equations, your goal is usually to get all terms involving the same angle and the same type of function (e.g., all in terms of \(\sin \theta\) or all in terms of \(\tan 2\theta\)).

3. Addition and Double Angle Formulae

These formulae allow you to expand and combine trigonometric expressions involving sums and multiples of angles.

3.1 The Addition Formulae (\(A \pm B\))

These formulae are given in the formula booklet (MF19), but knowing them well saves time.

• Sine: $$ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B $$ (The sign stays the same.)
• Cosine: $$ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B $$ (The sign flips!)
• Tangent: $$ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} $$ (The numerator sign stays; the denominator sign flips.)

Did you know? You can use these to find exact values for angles that aren't 30, 45, or 60, such as \(\cos 15^\circ\). Since \(15^\circ = 45^\circ - 30^\circ\), you can calculate \(\cos(45^\circ - 30^\circ)\) exactly!

3.2 The Double Angle Formulae (\(2A\))

These are derived directly from the addition formulae by setting \(A = B\).

• Sine: $$ \mathbf{\sin 2A = 2 \sin A \cos A} $$
• Tangent: $$ \mathbf{\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}} $$
• Cosine (The Triple Threat): \(\cos 2A\) has three forms, and you must know when to use each one effectively (especially in integration problems).
  1. $$ \mathbf{\cos 2A = \cos^2 A - \sin^2 A} $$
  2. $$ \mathbf{\cos 2A = 2 \cos^2 A - 1} $$ (Used if you only want \(\cos\))
  3. $$ \mathbf{\cos 2A = 1 - 2 \sin^2 A} $$ (Used if you only want \(\sin\))

Important Usage Tip: The double angle formulae aren't just for doubling; they are for changing the angle. If you see \(\sin 6x\), you can write it as \(2 \sin 3x \cos 3x\). If you see \(\cos^2 5x\), you can write it as \(\frac{1}{2}(1 + \cos 10x)\). This angle halving/doubling technique is often necessary when solving equations.

Key Takeaway: The Addition and Double Angle Formulae are essential tools for proofs and for unifying expressions so they contain only one type of angle.

4. The Harmonic Form (R-Formula)

The Harmonic Form, often called the R-formula, is used to convert an expression containing a combination of sine and cosine terms (like \(a \sin \theta + b \cos \theta\)) into a single trigonometric wave form, such as \(R \sin(\theta \pm \alpha)\) or \(R \cos(\theta \pm \alpha)\).

Analogy: Imagine combining two distinct sound waves into one clear, louder wave. The R-formula finds the amplitude (\(R\)) and the phase shift (\(\alpha\)) of the resulting combined wave.

4.1 The Goal and the Forms

You need to be able to express \(a \sin \theta + b \cos \theta\) into any of the four forms:

1. $$ R \sin(\theta + \alpha) $$
2. $$ R \sin(\theta - \alpha) $$
3. $$ R \cos(\theta + \alpha) $$
4. $$ R \cos(\theta - \alpha) $$

Where \(R > 0\) and \(0^\circ < \alpha < 90^\circ\) (or \(0 < \alpha < \pi/2\) radians).

4.2 Step-by-Step Calculation (Example: \(R \sin(\theta + \alpha)\))

Let's use the expression \(3 \sin \theta + 4 \cos \theta\). We want to write it in the form \(R \sin(\theta + \alpha)\).

Step 1: Expand the chosen R-form using the addition formula.

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

$$ 3 \sin \theta + 4 \cos \theta = (R \cos \alpha) \sin \theta + (R \sin \alpha) \cos \theta $$

Step 2: Compare coefficients of \(\sin \theta\) and \(\cos \theta\).

Comparing \(\sin \theta\) coefficients: $$ 3 = R \cos \alpha \quad \text{ (Equation 1)} $$ Comparing \(\cos \theta\) coefficients: $$ 4 = R \sin \alpha \quad \text{ (Equation 2)} $$

Step 3: Calculate R (the Amplitude).

Square both equations and add them:

$$ 3^2 + 4^2 = (R \cos \alpha)^2 + (R \sin \alpha)^2 $$

$$ 9 + 16 = R^2 (\cos^2 \alpha + \sin^2 \alpha) $$

Since \(\cos^2 \alpha + \sin^2 \alpha = 1\):

$$ 25 = R^2 \quad \implies \mathbf{R=5} $$

Step 4: Calculate \(\alpha\) (the Phase Shift).

Divide Equation 2 by Equation 1:

$$ \frac{R \sin \alpha}{R \cos \alpha} = \frac{4}{3} $$

$$ \tan \alpha = \frac{4}{3} $$

Using a calculator (and ensuring the correct mode, degrees or radians, as specified in the question):

$$ \mathbf{\alpha \approx 53.13^\circ} $$

Step 5: Write the final expression.

$$ \mathbf{3 \sin \theta + 4 \cos \theta = 5 \sin(\theta + 53.1^\circ)} \text{ (to 3 s.f.)} $$

4.3 Applications of the R-Formula

1. Maximum and Minimum Values:

Since \(-1 \leq \sin(\theta + \alpha) \leq 1\), the expression \(R \sin(\theta + \alpha)\) has:

• Maximum value: R (occurs when \(\sin(\theta + \alpha) = 1\))
• Minimum value: -R (occurs when \(\sin(\theta + \alpha) = -1\))

2. Solving Equations:

The R-formula converts a tricky equation like \(3 \sin \theta + 4 \cos \theta = 2\) into a solvable basic trigonometric equation:

$$ 5 \sin(\theta + 53.13^\circ) = 2 $$

$$ \sin(\theta + 53.13^\circ) = 0.4 $$

You then solve for the compound angle \((\theta + \alpha)\) and subtract \(\alpha\) to find \(\theta\). (Remember to adjust the range of \(\theta\) to the range of \((\theta + \alpha)\)!)

Common Mistake to Avoid: When using the expansion method (comparing coefficients), \(\alpha\) is always acute ($0^\circ < \alpha < 90^\circ$). Do not use the CAST diagram to find other possible values of \(\alpha\). The signs of $a$ and $b$ (the coefficients) inherently handle the quadrant of the final wave shift.

5. Solving Advanced Trigonometric Equations

This is where all the tools come together. You must be precise about the required interval and the function you are solving for.

5.1 Standard Procedure for Solving

1. Unify the Expression: Use identities (Sections 2 and 3) to convert the equation so that it involves only:

• One trigonometric function (e.g., all \(\tan x\)), OR
• One type of angle (e.g., all \(\theta\) or all \(2\theta\)), OR
• The Harmonic Form (\(R\sin(\theta + \alpha)\)).

2. Adjust the Range: If you are solving for a modified angle (e.g., \(2\theta\), or \(\theta + 30^\circ\)), make sure you adjust the given domain (interval) accordingly.

Example: If \(0^\circ \leq \theta \leq 360^\circ\), then \(0^\circ \leq 2\theta \leq 720^\circ\).

3. Find the Principal Value (PV): Calculate the basic angle using the inverse function (e.g., \(\sin^{-1}, \cos^{-1}\)). Ignore the sign of the value at this stage.

4. Find All Solutions using CAST: Use the CAST diagram (or the graph) to find all angles within the adjusted range that correspond to the sign of the original ratio.

5. Final Step: Adjust your solutions back to the original variable (\(\theta\)).

5.2 Common Tricky Scenarios

Scenario 1: Using Double Angles to Solve

If you have a mix of angles like \(\sin 2\theta\) and \(\cos \theta\), you must convert the double angle:

$$ \sin 2\theta - \cos \theta = 0 $$ $$ 2 \sin \theta \cos \theta - \cos \theta = 0 $$ $$ \cos \theta (2 \sin \theta - 1) = 0 $$

This gives two separate, simpler equations to solve: \(\cos \theta = 0\) and \(\sin \theta = 1/2\).

Scenario 2: Equations involving reciprocal functions

If you encounter something like \(\tan \theta + \cot \theta = 4\):

$$ \tan \theta + \frac{1}{\tan \theta} = 4 $$

Multiply by \(\tan \theta\) to get a quadratic in \(\tan \theta\):

$$ \tan^2 \theta - 4 \tan \theta + 1 = 0 $$

You can then solve using the quadratic formula, giving two basic values for \(\tan \theta\) that must be solved using the standard procedure.

5.3 The Importance of Radians

Many P3 questions will specify the interval in radians (e.g., \(-\pi < x < \pi\) or \(0 \leq x \leq 2\pi\)).

• Always ensure your calculator is in Radian Mode.
• Remember key values: \(\pi \approx 3.142\). \(2\pi \approx 6.283\).
• If you are working with the R-formula, calculate \(\alpha\) in radians from the start.

Key Takeaway: Trigonometric problems in P3 are highly procedural. Identify the required identity or formula, unify the expression, adjust the range, and systematically find all solutions using the CAST method.