A Comprehensive Guide to Additional Mathematics Trigonometry (0606)

Hello future Add Math expert! Welcome to the world of Trigonometry. While you've met sine, cosine, and tangent before, Additional Mathematics takes these concepts further—you'll explore their graphs, use powerful identities, and solve much more complex equations.

Why is this important? Trigonometry is the foundation for topics like waves, harmonic motion, and advanced calculus. Mastering this chapter gives you the tools needed to tackle sophisticated problems both on Paper 1 and Paper 2. Let’s dive in!

Section 1: The Six Trigonometric Functions (10.1)

1.1 Recalling the Basics: SOH CAH TOA

In your IGCSE Mathematics course, you mainly used the primary ratios based on a right-angled triangle (for acute angles):

  • Sine (\(\sin \theta\)): Opposite / Hypotenuse
  • Cosine (\(\cos \theta\)): Adjacent / Hypotenuse
  • Tangent (\(\tan \theta\)): Opposite / Adjacent

1.2 Introducing the Reciprocal Functions

In Add Math, we use three additional functions, which are simply the reciprocals of the primary ones. Don't worry, they are easy to remember once you get the pattern!

  • Cosecant (\(\csc \theta\) or \(\cosec \theta\)): The reciprocal of Sine.
    \(\cosec \theta = \frac{1}{\sin \theta}\)
  • Secant (\(\sec \theta\)): The reciprocal of Cosine.
    \(\sec \theta = \frac{1}{\cos \theta}\)
  • Cotangent (\(\cot \theta\)): The reciprocal of Tangent.
    \(\cot \theta = \frac{1}{\tan \theta}\)

Memory Trick: To remember which reciprocal goes with which primary function, notice that the third letter of the reciprocal function tells you which primary function it pairs with: Cosecant pairs with Sine; Secant pairs with Cosine.

1.3 Angles of Any Magnitude (The CAST Diagram)

Trigonometry in Additional Mathematics requires us to find the ratio of angles larger than \(90^\circ\). We measure angles anticlockwise from the positive x-axis.

The CAST diagram helps us determine which functions are positive in which quadrant:

  • C (Quadrant IV, \(270^\circ < \theta < 360^\circ\)): Only Cosine (and its reciprocal, Secant) is positive.
  • A (Quadrant I, \(0^\circ < \theta < 90^\circ\)): All functions are positive.
  • S (Quadrant II, \(90^\circ < \theta < 180^\circ\)): Only Sine (and its reciprocal, Cosecant) is positive.
  • T (Quadrant III, \(180^\circ < \theta < 270^\circ\)): Only Tangent (and its reciprocal, Cotangent) is positive.

Step-by-Step for Solving Any Magnitude Angle:

  1. Find the Basic Angle (\(\alpha\)): Use the positive value of the ratio (e.g., if \(\sin \theta = -0.5\), use \(\sin \alpha = 0.5\)). This angle \(\alpha\) will always be acute (\(0^\circ < \alpha < 90^\circ\)).
  2. Identify the Quadrants: Use the CAST rule based on whether the original ratio was positive or negative.
  3. Calculate the Angles (\(\theta\)):
    • Quadrant I: \(\theta = \alpha\)
    • Quadrant II: \(\theta = 180^\circ - \alpha\)
    • Quadrant III: \(\theta = 180^\circ + \alpha\)
    • Quadrant IV: \(\theta = 360^\circ - \alpha\)

Key Takeaway: The six functions are defined by their reciprocals. The CAST diagram is essential for finding angles beyond the first quadrant.

Section 2: Graphing Trigonometric Functions (10.2 & 10.3)

Understanding the graphs of trigonometric functions is key to understanding amplitude, period, and solving inequalities graphically.

2.1 Amplitude, Period, and Vertical Shift

We focus on transformations of the standard forms:
\(\mathbf{y = a \sin bx + c}\)
\(\mathbf{y = a \cos bx + c}\)
\(\mathbf{y = a \tan bx + c}\)

a) Amplitude (\(a\))

The value \(a\) (which the syllabus specifies is a positive integer) determines the amplitude (vertical stretch).

  • For sine and cosine graphs, the amplitude is \(|a|\). It is the distance from the centre line (\(y=c\)) to the maximum or minimum point.
  • The range of the function is \([c - a, c + a]\).
b) Period (\(b\))

The value \(b\) (a simple fraction or integer) determines the period (horizontal compression or stretch). This is the length of one complete cycle.

  • For Sine and Cosine: Period \(= \frac{360^\circ}{b}\) (in degrees) or \(= \frac{2\pi}{b}\) (in radians).
  • For Tangent: Period \(= \frac{180^\circ}{b}\) (in degrees) or \(= \frac{\pi}{b}\) (in radians). (Tangents naturally repeat twice as fast as sine/cosine).

Example: For \(y = 3 \cos 2x\), the amplitude is 3, and the period is \(360^\circ / 2 = 180^\circ\). The graph completes two cycles where the standard cosine graph completes one.

c) Vertical Shift (\(c\))

The value \(c\) (an integer) shifts the entire graph up or down. This becomes the midline or central axis of the sine/cosine wave.

2.2 Sketching Graphs (10.3)

When sketching, always clearly label:

  1. The y-intercept (set \(x=0\)).
  2. The maximum and minimum values.
  3. The x-intercepts (if within the given domain).
  4. If sketching a tangent graph, the asymptotes must be labelled with their equations (e.g., \(x = 90^\circ\)).

Tangent Graphs and Asymptotes:
Since \(\tan x = \frac{\sin x}{\cos x}\), the function is undefined whenever \(\cos x = 0\). These vertical lines are called asymptotes.

  • Standard \(\tan x\) asymptotes occur at \(x = 90^\circ, 270^\circ, -90^\circ,\) etc.
  • For \(y = a \tan bx + c\), you find the asymptotes by solving \(bx = 90^\circ + n(180^\circ)\).

Did you know? The concept of period is crucial in physics! It describes the time taken for waves (like sound or light) to complete one cycle.

Key Takeaway: \(a\) affects height (amplitude), \(b\) affects width (period), and \(c\) affects position (vertical shift). Know how to calculate the period for sine/cosine versus tangent.

Section 3: Essential Trigonometric Identities (10.4)

Identities are equations that are true for all values of the variable. You need to know how to use these three foundational Pythagorean Identities (which are provided in the formula sheet, but practice using them):

3.1 The Primary Identity

This comes directly from Pythagoras' Theorem on a unit circle:
\(\sin^2 A + \cos^2 A = 1\)

You often need to rearrange this identity, so practice expressing \(\sin^2 A\) as \(1 - \cos^2 A\), or vice versa.

3.2 The Reciprocal Identities

These two identities are derived by dividing the primary identity (\(\sin^2 A + \cos^2 A = 1\)) by either \(\cos^2 A\) or \(\sin^2 A\).


Identity 2 (Dividing by \(\cos^2 A\)):
\(\frac{\sin^2 A}{\cos^2 A} + \frac{\cos^2 A}{\cos^2 A} = \frac{1}{\cos^2 A}\)
\(\tan^2 A + 1 = \sec^2 A\)


Identity 3 (Dividing by \(\sin^2 A\)):
\(\frac{\sin^2 A}{\sin^2 A} + \frac{\cos^2 A}{\sin^2 A} = \frac{1}{\sin^2 A}\)
\(1 + \cot^2 A = \cosec^2 A\)

Quick Review: Identities to Memorise (or Locate Quickly)
  • \(\sin^2 A + \cos^2 A = 1\)
  • \(\sec^2 A = 1 + \tan^2 A\)
  • \(\cosec^2 A = 1 + \cot^2 A\)

Key Takeaway: These identities allow you to swap between different trigonometric functions (e.g., changing a cosine equation into a sine equation, or changing a secant equation into a tangent equation).

Section 4: Solving Trigonometric Equations (10.5)

Solving trigonometric equations is where you combine all your skills: basic angles, CAST rule, and algebraic manipulation using identities.

4.1 General Solving Strategy

Most complex Add Math equations will look like a quadratic equation once an identity has been used.

Step-by-Step Example (Quadratics in Trig): Solve \(2 \sec^2 \theta + 3 \tan \theta - 5 = 0\) for \(0^\circ \le \theta \le 360^\circ\).

  1. Homogenise the equation: The equation has \(\sec^2 \theta\) and \(\tan \theta\). We must make them the same function using an identity. Use \(\sec^2 \theta = 1 + \tan^2 \theta\).
    (We choose to convert secant to tangent because converting tangent to secant involves square roots, which is messier.)
    \(2(1 + \tan^2 \theta) + 3 \tan \theta - 5 = 0\)
  2. Simplify to a Quadratic:
    \(2 + 2 \tan^2 \theta + 3 \tan \theta - 5 = 0\)
    \(2 \tan^2 \theta + 3 \tan \theta - 3 = 0\)
  3. Solve the Quadratic: Let \(x = \tan \theta\).
    \(2x^2 + 3x - 3 = 0\)
    Since this doesn't factorise easily, use the quadratic formula (Paper 2) or factorisation (Paper 1, if it allows).
    \(x = \frac{-3 \pm \sqrt{9 - 4(2)(-3)}}{4} = \frac{-3 \pm \sqrt{33}}{4}\)
    This gives two values for \(\tan \theta\):
    \(\tan \theta \approx 0.686\) OR \(\tan \theta \approx -2.186\)
  4. Find the Basic Angles (\(\alpha\)) and Use CAST:
    • Case 1: \(\tan \theta = 0.686\) (Positive, Quadrants I & III)
      \(\alpha = \tan^{-1}(0.686) \approx 34.4^\circ\)
      \(\theta_1 = 34.4^\circ\) (Q I)
      \(\theta_2 = 180^\circ + 34.4^\circ = 214.4^\circ\) (Q III)
    • Case 2: \(\tan \theta = -2.186\) (Negative, Quadrants II & IV)
      \(\alpha = \tan^{-1}(2.186) \approx 65.4^\circ\)
      \(\theta_3 = 180^\circ - 65.4^\circ = 114.6^\circ\) (Q II)
      \(\theta_4 = 360^\circ - 65.4^\circ = 294.6^\circ\) (Q IV)

The solutions are \(34.4^\circ, 114.6^\circ, 214.4^\circ, 294.6^\circ\) (to 1 d.p.).

4.2 Common Mistakes to Avoid

  • Dividing by a Variable: NEVER divide an equation by \(\sin \theta\) or \(\cos \theta\). If you do, you lose solutions where \(\sin \theta = 0\) or \(\cos \theta = 0\). Always factorise instead (e.g., solve \(\sin \theta \cos \theta = \sin \theta\) by writing \(\sin \theta \cos \theta - \sin \theta = 0\) and factorising \(\sin \theta (\cos \theta - 1) = 0\)).
  • Angle in the Ratio: If the angle is transformed (e.g., \(2\theta\) or \(\theta/3\)), remember to adjust the domain first before listing solutions.
    Example: If the domain is \(0^\circ \le \theta \le 360^\circ\), then the domain for \(2\theta\) is \(0^\circ \le 2\theta \le 720^\circ\). You must find all solutions up to \(720^\circ\) before dividing by 2.
  • Reciprocal Calculations: If you have \(\sec \theta = 2\), remember to solve \(\cos \theta = 1/2\), not \(\cos \theta = 2\)!

Key Takeaway: Convert the equation into a single trigonometric function using identities, then solve it as a quadratic equation. Always check the domain carefully.

Section 5: Proving Trigonometric Relationships (10.6)

Proving identities means showing that one side of the equation is algebraically equivalent to the other side. You cannot move terms across the equals sign!

5.1 Strategies for Proofs

  1. Start with the More Complex Side: It is usually easier to simplify a complex expression than to complicate a simple one.
  2. Convert to Sine and Cosine: If you see \(\sec\), \(\csc\), \(\cot\), or \(\tan\), your first step should often be rewriting them using their definitions in terms of \(\sin\) and \(\cos\).
    Example: Replace \(\cot x\) with \(\frac{\cos x}{\sin x}\) and \(\sec x\) with \(\frac{1}{\cos x}\).
  3. Look for Pythagorean Opportunities: If you see a squared term (\(\sin^2 x\), \(\cos^2 x\), etc.), check if the primary identities (\(\sin^2 A + \cos^2 A = 1\), etc.) can be applied immediately.
  4. Combine Fractions: If you have two fractions, find a common denominator to combine them.
  5. Factorisation: Look out for differences of squares (e.g., \(1 - \cos^2 A\) can be written as \((1 - \cos A)(1 + \cos A)\)) or common factors.
Example of a Proof Strategy

Prove: \(\sin x \tan x + \cos x = \sec x\)

Start with the Left Hand Side (LHS), as it is more complex:
LHS: \(\sin x \tan x + \cos x\)
Step 1: Convert \(\tan x\) to \(\frac{\sin x}{\cos x}\).
LHS: \(\sin x \left( \frac{\sin x}{\cos x} \right) + \cos x\)
LHS: \(\frac{\sin^2 x}{\cos x} + \cos x\)
Step 2: Combine the fractions (Common denominator is \(\cos x\)).
LHS: \(\frac{\sin^2 x}{\cos x} + \frac{\cos^2 x}{\cos x}\)
LHS: \(\frac{\sin^2 x + \cos^2 x}{\cos x}\)
Step 3: Use the identity \(\sin^2 x + \cos^2 x = 1\).
LHS: \(\frac{1}{\cos x}\)
Step 4: Use the reciprocal definition.
LHS: \(\sec x\)
Since LHS = RHS, the identity is proved.

Don't worry if this seems tricky at first—proofs require creativity and lots of practice! The more you use the identities, the faster you will see the patterns.

Key Takeaway: When proving identities, work only on one side (usually the complicated one) and convert everything to sine and cosine initially if you get stuck.