Pure Mathematics | Trigonometry

Unit P1: Pure Mathematics 1 - Comprehensive Study Notes: Trigonometry

Hello Future Mathematician! Welcome to Trigonometry!

Welcome to one of the most practical and beautiful chapters in Pure Mathematics! If you’ve encountered SOH CAH TOA before, that was just the beginning. In P1 Trigonometry, we move beyond right-angled triangles to look at angles of any size (big, small, and even negative!).

Trigonometry is essential for understanding cyclical patterns (like waves, sound, and orbits). Don't worry if this seems tricky at first; we will break down the graphs, formulas, and solving techniques step-by-step. Let's get started!

1. Measuring Angles: Degrees and Radians

1.1 Review: The Basics of Angles (Prerequisite)

Remember that a full rotation is \(360^\circ\). We measure angles starting from the positive x-axis (the initial line) and rotating anti-clockwise.

1.2 Introducing Radian Measure

In advanced mathematics, using degrees can be cumbersome. Instead, we use radians. Radian measure is the "natural" way to measure angles.

What is a Radian?

A radian is defined as the angle subtended at the centre of a circle when the arc length is equal to the radius.

Analogy: Imagine a piece of string exactly the length of the radius. If you bend that string around the edge of the circle, the angle it cuts off is 1 radian.

Key Conversion: Degrees to Radians

Since the circumference of a circle is \(2\pi r\), there are \(2\pi\) radians in a full circle.
Therefore, the fundamental relationship is:
\(\pi \text{ radians} = 180^\circ\)

To convert:

• Degrees to Radians: Multiply by \(\frac{\pi}{180}\)
• Radians to Degrees: Multiply by \(\frac{180}{\pi}\)

Example Conversions:
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2. The Unit Circle and the CAST Diagram

To deal with angles greater than \(90^\circ\) or negative angles, we use the Unit Circle.

2.1 Defining Ratios on the Unit Circle

The Unit Circle is a circle centred at the origin (0, 0) with a radius of 1.

If a line rotates by angle \(\theta\) and meets the unit circle at point P(x, y):

• \(\cos \theta\) is the x-coordinate of P.
• \(\sin \theta\) is the y-coordinate of P.
• \(\tan \theta\) is the slope (\(\frac{y}{x}\)) of the line OP.

2.2 Understanding the CAST Diagram (Signs of Ratios)

The CAST diagram helps us remember which trigonometric ratios are positive in each of the four quadrants.

(Imagine a standard Cartesian plane split into four quadrants.)

• Quadrant I (0° to 90°): All. Sine, Cosine, and Tangent are ALL positive. (x > 0, y > 0)
• Quadrant II (90° to 180°): Sine. Only Sine is positive. (x < 0, y > 0)
• Quadrant III (180° to 270°): Tangent. Only Tangent is positive. (x < 0, y < 0, so \(\frac{y}{x} > 0\))
• Quadrant IV (270° to 360°): Cosine. Only Cosine is positive. (x > 0, y < 0)

Memory Aid (Mnemonic):

Start from Quadrant IV (bottom right) and go anti-clockwise:
Cats Always Sit Together.
(Or, Central All Students Take...)

2.3 Finding Related Angles (Reference Angle)

The reference angle (\(\alpha\)) is the acute angle that the terminal arm makes with the horizontal axis (the x-axis).

If \(\theta\) is the angle measured from the positive x-axis:

• QI: \(\theta = \alpha\)
• QII: \(\theta = 180^\circ - \alpha\) (or \(\pi - \alpha\))
• QIII: \(\theta = 180^\circ + \alpha\) (or \(\pi + \alpha\))
• QIV: \(\theta = 360^\circ - \alpha\) (or \(2\pi - \alpha\))

This technique is essential for solving equations later!

3. Trigonometric Graphs and Periodicity

The graph of a trigonometric function shows how the ratio changes as the angle (\(\theta\) or \(x\)) increases.

3.1 The Sine Graph: \(y = \sin x\)

• Shape: Smooth wave, starting at (0, 0).
• Period: \(360^\circ\) (or \(2\pi\) radians). The wave repeats every \(360^\circ\).
• Range: \(-1 \le y \le 1\). The amplitude is 1.
• Key Points (in degrees): \(0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ\) correspond to \(0, 1, 0, -1, 0\).

3.2 The Cosine Graph: \(y = \cos x\)

• Shape: Smooth wave, starting at (0, 1).
• Relationship: The cosine graph is simply the sine graph shifted \(90^\circ\) (or \(\frac{\pi}{2}\)) to the left.
• Period: \(360^\circ\) (or \(2\pi\) radians).
• Range: \(-1 \le y \le 1\).
• Key Points (in degrees): \(0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ\) correspond to \(1, 0, -1, 0, 1\).

3.3 The Tangent Graph: \(y = \tan x\)

The tangent graph is very different because \(\tan x = \frac{\sin x}{\cos x}\).

• Period: \(180^\circ\) (or \(\pi\) radians). It repeats much faster than sine/cosine.
• Range: All real numbers (\(y \in \mathbb{R}\)).
• Asymptotes: Since division by zero is undefined, the tangent graph has vertical asymptotes whenever \(\cos x = 0\).
• Locations of Asymptotes: \(x = 90^\circ, 270^\circ, -90^\circ, \text{ etc.}\) (or \(\frac{\pi}{2}, \frac{3\pi}{2}\)).

4. The Fundamental Trigonometric Identities (P1)

Identities are equations that are true for all values of the variable. You must know these two identities perfectly for P1. They are used to simplify complex expressions or prove other identities.

4.1 Identity 1: The Ratio Identity

This identity comes directly from the definition of the unit circle, where \(\tan \theta\) is the gradient (rise over run, or y over x).

\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

This is true provided \(\cos \theta \neq 0\).

4.2 Identity 2: The Pythagorean Identity

This identity is derived using Pythagoras' Theorem on the unit circle.
Consider the right triangle formed by the point P(x, y) and the origin. Since the radius (hypotenuse) is 1, and \(x = \cos\theta\) and \(y = \sin\theta\):
\(x^2 + y^2 = 1^2\)

\(\sin^2 \theta + \cos^2 \theta = 1\)

Important Notes on Notation:

\(\sin^2 \theta\) means \((\sin \theta)^2\). It does not mean \(\sin (\theta^2)\). This is a very common source of error!

Applying the Identities (Example):

If you are given that \(\sin \theta = 0.6\), and you need to find \(\cos \theta\):
We use the Pythagorean identity:
\((0.6)^2 + \cos^2 \theta = 1\)
\(0.36 + \cos^2 \theta = 1\)
\(\cos^2 \theta = 0.64\)
\(\cos \theta = \pm 0.8\)
(The sign (\(+\) or \(-\)) depends entirely on the quadrant \(\theta\) is in!)

5. Solving Basic Trigonometric Equations

Solving trig equations means finding all the angle values that satisfy the equation within a given range (e.g., \(0^\circ \le x < 360^\circ\)).

5.1 The Four Step Process for Solving Equations

Step 1: Isolate the Trigonometric Ratio

Treat the trig function (e.g., \(\sin x\)) like a variable (like \(y\)) and rearrange the equation.
Example: Solve \(2\cos x - 1 = 0\).
\(2\cos x = 1 \implies \cos x = 0.5\)

Step 2: Find the Principal Value (PV) / Reference Angle (\(\alpha\))

Use the inverse function on your calculator (e.g., \(\arccos\), \(\arcsin\), \(\arctan\)) on the positive value of the ratio to find the acute angle \(\alpha\). This is your reference angle.
Example: For \(\cos x = 0.5\).
\(\alpha = \arccos(0.5) = 60^\circ\).

Step 3: Determine the Correct Quadrants using CAST

Look at the sign of the ratio in Step 1 to decide which quadrants contain solutions.
Example: \(\cos x\) is positive (0.5). Cosine is positive in Quadrant I (A) and Quadrant IV (C).

Step 4: Calculate the Solutions in the Given Range

Use the reference angle \(\alpha\) and the quadrant formulas (from Section 2.3) to find the final angles \(\theta\).

Example (Range \(0^\circ \le x < 360^\circ\)):

• Q I Solution: \(x = \alpha = 60^\circ\)
• Q IV Solution: \(x = 360^\circ - \alpha = 360^\circ - 60^\circ = 300^\circ\)

5.2 Handling Equations with Transformations (Brief Introduction)

Sometimes you will solve equations involving \(f(kx)\), for example, \(\sin(2x) = 0.5\).

The key trick here is to adjust the range before you start solving.
If the range is \(0^\circ \le x < 360^\circ\), then the range for \(2x\) is \(0^\circ \le 2x < 720^\circ\).

Step-by-step for \(2x\):

1. Let \(\theta = 2x\). Adjust the range for \(\theta\).
2. Solve \(\sin \theta = 0.5\) normally, finding all solutions for \(\theta\) within the expanded range. (You will likely find four solutions, repeating the process beyond \(360^\circ\)).
3. Divide all your solutions for \(\theta\) by 2 to find the final solutions for \(x\).

You've got this! Trigonometry links geometry, algebra, and graphical analysis, making it a powerful tool. Practice the CAST diagram method until it becomes second nature.