P2.4 Trigonometry: Comprehensive Study Notes (9660)

Welcome to the world of advanced Trigonometry! In AS Level, you learned the fundamental rules and functions. In P2, we dive deeper into identities, reciprocal functions, and powerful formulae that allow us to combine, simplify, and solve much more complex trigonometric problems. This is essential knowledge, especially as these identities are heavily used later in Differentiation and Integration.

1. The Foundations: Functions and Key Identities (AS Review)

Before moving to P2 material, make sure you are fluent with the standard sine, cosine, and tangent functions.

Basic Definitions and Graphs

The core functions are periodic, meaning their graphs repeat over a fixed interval.

  • Sine (\(\sin x\)): Period \(360^\circ\) or \(2\pi\) rad. Range \([-1, 1]\). Starts at (0, 0).
  • Cosine (\(\cos x\)): Period \(360^\circ\) or \(2\pi\) rad. Range \([-1, 1]\). Starts at (0, 1).
  • Tangent (\(\tan x\)): Period \(180^\circ\) or \(\pi\) rad. Range \( ( -\infty, \infty ) \). Has vertical asymptotes at \(90^\circ, 270^\circ, \dots\).
Essential Identities (Must Know by Heart)

These two identities form the bedrock of almost all trigonometric proofs and simplifications:

  1. The Quotient Identity:

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

  2. The Pythagorean Identity:

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

    Memory Aid: This identity comes directly from Pythagoras' Theorem applied to a right-angled triangle in the unit circle: \((\text{opposite})^2 + (\text{adjacent})^2 = (\text{hypotenuse})^2\), which is \((\sin\theta)^2 + (\cos\theta)^2 = 1^2\).

Quick Review: Solving Simple Equations

Remember the critical four-step process for solving equations in a given range (e.g., \(0^\circ \le x \le 360^\circ\)):
1. Isolate the trigonometric function.
2. Find the Principal Value (PV) using \(\sin^{-1}, \cos^{-1}\), or \(\tan^{-1}\) (always positive, acute angle).
3. Use the CAST diagram or function graphs to find all valid quadrants.
4. Calculate the angles within the required range. (Don't forget to check both ends of the range!)

Key Takeaway for Section 1: Master the three basic graphs and be able to use the fundamental Pythagorean and Quotient identities instantly.

2. Radian Measure, Arc Length, and Sector Area

Radians are the standard unit of angle measure in A-Level maths (especially calculus). When no unit is specified, assume radians.

Understanding Radians

A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. It's a fundamental unit, unlike degrees, which are arbitrary.

  • Conversion: \( \pi \text{ radians} = 180^\circ \)
  • To convert Degrees to Radians, multiply by \( \frac{\pi}{180} \).
  • To convert Radians to Degrees, multiply by \( \frac{180}{\pi} \).

Did you know? \( 1\) radian is approximately \(57.3^\circ\).

Arc Length and Sector Area Formulae

These formulae are only valid when the angle \(\theta\) is measured in radians.

  • Arc Length (\(l\)): $$ l = r\theta $$
  • Area of a Sector (\(A\)): $$ A = \frac{1}{2}r^2\theta $$

Analogy: These formulas are much simpler than their degree counterparts (which involve multiplying by \(\frac{\theta}{360} \times 2\pi r\) or \(\frac{\theta}{360} \times \pi r^2\)). Using radians is efficient!

Key Takeaway for Section 2: If the question involves arc length or sector area, or if it's a P2 calculus problem, ensure your calculator is set to RADIAN MODE and use the \(l=r\theta\) and \(A=\frac{1}{2}r^2\theta\) formulae.

3. Advanced Trigonometric Functions and Identities (P2 Content)

In P2, we introduce three new functions—the reciprocals of the original three—and three powerful new identities derived from the basic Pythagorean identity.

Reciprocal Functions

These functions are simply the multiplicative inverses (flips) of sine, cosine, and tangent:

  • Secant (\(\sec x\)): \( \sec x = \frac{1}{\cos x} \)
  • Cosecant (\(\operatorname{cosec} x\) or \(\csc x\)): \( \operatorname{cosec} x = \frac{1}{\sin x} \)
  • Cotangent (\(\cot x\)): \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \)

Common Mistake to Avoid: Students often mix up secant and cosecant. The trick is to look at the third letter: 'S'ecant goes with 'C'osine; 'C'osecant goes with 'S'ine.

Derived Pythagorean Identities

We can generate two more fundamental identities by dividing the original identity (\( \sin^2\theta + \cos^2\theta = 1 \)) by either \(\cos^2\theta\) or \(\sin^2\theta\).

Identity 1: Dividing by \(\cos^2\theta\)

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

Which simplifies to:

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

Identity 2: Dividing by \(\sin^2\theta\)

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

Which simplifies to:

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

Memory Aid: To remember which goes with which, think of pairs that use the same "first letter":

  • Secant goes with Tangent (\(\sec^2\theta\) and \(\tan^2\theta\))
  • Cosecant goes with Cotangent (\(\operatorname{cosec}^2\theta\) and \(\cot^2\theta\))

Inverse Trigonometric Functions (\(\sin^{-1}, \cos^{-1}, \tan^{-1}\))

The inverse functions are used to find the angle that corresponds to a given ratio. Because the original trig functions are periodic (many angles give the same output), their inverse functions must have restricted domains and ranges to ensure they are true functions.

The output of an inverse function is called the principal value. You need to know these ranges:

  • \(y = \sin^{-1} x\): Domain \([-1, 1]\). Range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) or \([-90^\circ, 90^\circ]\).
  • \(y = \cos^{-1} x\): Domain \([-1, 1]\). Range \( [0, \pi] \) or \([0^\circ, 180^\circ]\).
  • \(y = \tan^{-1} x\): Domain \( (-\infty, \infty) \). Range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) or \((-90^\circ, 90^\circ)\).

Analogy: Imagine a cinema screen showing a repeating movie (the trig graph). The inverse function only shows you a small, unique clip (the principal range) that contains all possible height values once.

Key Takeaway for Section 3: The three new reciprocal functions and the two derived Pythagorean identities (\(1+\tan^2\theta=\sec^2\theta\) and \(1+\cot^2\theta=\operatorname{cosec}^2\theta\)) are vital tools for proofs and simplification. Remember the principal ranges for inverse functions.

4. Compound Angle Formulae (Addition Formulae)

These formulae allow you to expand trigonometric functions of the sum or difference of two angles, like \(\sin(A+B)\).

The Formulae (Provided in the Formula Booklet)
  • $$ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B $$
  • $$ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B $$
  • $$ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} $$

Memory Aid:

  • Sine is "Mixed" and "Keeps the sign": Sine, Cosine, Cosine, Sine.
  • Cosine is "Pure" and "Changes the sign": Cosine, Cosine, Sine, Sine.

Application: You can use these to find exact values for angles that aren't standard, like finding the exact value of \(\cos(75^\circ)\):

Step-by-step Example: 1. Recognise \(75^\circ = 45^\circ + 30^\circ\). 2. Use the formula: \( \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \) 3. Substitute exact values: \( \left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right) \) 4. Simplify: \( \frac{\sqrt{3} - 1}{2\sqrt{2}} \)

5. Double Angle Formulae

These are a special case of the compound angle formulae, where \(A=B\). They are critical for integrating expressions involving \(\sin^2 x\) or \(\cos^2 x\).

The Formulae (Must Learn for P2)
  1. Sine Double Angle: $$ \sin 2A = 2 \sin A \cos A $$
  2. Cosine Double Angle (Three Forms):

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

    This is the fundamental form. By substituting \( \sin^2 A = 1 - \cos^2 A \) or \( \cos^2 A = 1 - \sin^2 A \), we get the other two:

    $$ \cos 2A = 2 \cos^2 A - 1 \quad \text{(Useful when you only want cosine terms)} $$

    $$ \cos 2A = 1 - 2 \sin^2 A \quad \text{(Useful when you only want sine terms)} $$

  3. Tangent Double Angle: $$ \tan 2A = \frac{2 \tan A}{1 - \tan^2 A} $$
Using Double Angle Identities to Simplify Equations

Often, trigonometric equations involve different angles (like \(2x\) and \(x\)) or different powers (\(\sin^2 x\) and \(\cos x\)). You must use these identities to rewrite the equation so that all terms have the same angle and the same function type (e.g., all terms in terms of \(\cos x\)).

Example: Solve \( 3\sin 2x = \cos x \) for \( 0 \le x < 2\pi \). 1. Use \( \sin 2x = 2\sin x \cos x \): $$ 3(2 \sin x \cos x) = \cos x $$ $$ 6 \sin x \cos x - \cos x = 0 $$ 2. Factorise (do NOT divide by \(\cos x\), or you lose solutions!): $$ \cos x (6 \sin x - 1) = 0 $$ 3. Set each factor to zero: $$ \cos x = 0 \quad \text{or} \quad 6 \sin x = 1 \implies \sin x = 1/6 $$ (Solve both resulting simple equations in radians.)

Key Takeaway for Section 4 & 5: The Addition and Double Angle formulae allow for complex simplification. Remember the three forms of \(\cos 2A\). These are essential not only for solving equations but for integration questions (e.g., if asked to integrate \(\cos^2 x\), you must first rewrite it as \(\frac{1}{2}(1 + \cos 2x)\)).

6. The R-Formula (Auxiliary Angle Form)

The R-formula (or Auxiliary Angle) is a powerful technique used to combine expressions of the form \( a\cos\theta + b\sin\theta \) into a single, more manageable trigonometric function: \( R\sin(\theta \pm \alpha) \) or \( R\cos(\theta \pm \alpha) \).

Why is the R-Formula useful?

If you have a function like \( f(\theta) = 3\sin\theta + 4\cos\theta \), it's hard to find the maximum or solve \( f(\theta) = 2 \). By converting it to a single function (e.g., \( 5\sin(\theta + 53.1^\circ) \)), these problems become easy.

Applications:

  • Finding the maximum and minimum values of an expression.
  • Finding the angle at which these extrema occur.
  • Solving complex equations.

The Process: Writing \( a\cos\theta + b\sin\theta \) in the form \( R\sin(\theta + \alpha) \)

We want \( a\cos\theta + b\sin\theta \equiv R\sin(\theta + \alpha) \). (The approach is similar for cosine forms.)

Step 1: Expand the Target Form $$ R\sin(\theta + \alpha) = R(\sin\theta \cos\alpha + \cos\theta \sin\alpha) $$ $$ R\sin(\theta + \alpha) = (R\cos\alpha)\sin\theta + (R\sin\alpha)\cos\theta $$

Step 2: Compare Coefficients Compare this expanded form to \( a\cos\theta + b\sin\theta \):

Coefficients of \(\sin\theta\): \( b = R\cos\alpha \quad \mathbf{(1)} \)

Coefficients of \(\cos\theta\): \( a = R\sin\alpha \quad \mathbf{(2)} \)

Step 3: Find R (The Amplitude) Square and add (1) and (2): $$ b^2 + a^2 = (R\cos\alpha)^2 + (R\sin\alpha)^2 $$ $$ a^2 + b^2 = R^2 (\cos^2\alpha + \sin^2\alpha) $$ Since \( \cos^2\alpha + \sin^2\alpha = 1 \): $$ R = \sqrt{a^2 + b^2} $$ R is always positive.

Step 4: Find \(\alpha\) (The Phase Shift) Divide (2) by (1): $$ \frac{R\sin\alpha}{R\cos\alpha} = \frac{a}{b} $$ $$ \tan\alpha = \frac{a}{b} $$ Calculate \( \alpha = \tan^{-1}\left(\frac{a}{b}\right) \). Since \(\alpha\) represents a phase shift, it is usually required to be acute, \( 0^\circ < \alpha < 90^\circ \) (or \( 0 < \alpha < \frac{\pi}{2} \)).

Common Mistake to Avoid: Make sure you match the coefficients correctly based on the formula you choose. For example, if using \( R\cos(\theta - \alpha) \), the coefficients derived will be different!

Example of Finding Extrema

If you found that \( 3\sin\theta + 4\cos\theta = 5\sin(\theta + 53.1^\circ) \):

  • The maximum value is \( R = 5 \), which occurs when \(\sin(\theta + 53.1^\circ) = 1\).
  • The minimum value is \( -R = -5 \), which occurs when \(\sin(\theta + 53.1^\circ) = -1\).
Key Takeaway for Section 6: The R-Formula is an algebraic conversion that turns two separate functions into one wave. Remember the key steps: 1. Expand. 2. Compare coefficients. 3. Find R using Pythagoras. 4. Find \(\alpha\) using Tangent.