Welcome to the World of Trigonometry!
Welcome! In this chapter, we are moving beyond simple right-angled triangles and diving into the beautiful, repeating patterns of Trigonometry. This is one of the most important chapters in AS Level Pure Mathematics because it connects geometry, algebra, and graphs. Whether you’re interested in engineering, music production (sound waves!), or architecture, trigonometry is the "mathematical glue" that holds it all together.
Don't worry if you found "SOH CAH TOA" a bit confusing in the past. We are going to break everything down into simple, manageable steps. Let’s get started!
1. Triangle Rules: Sine, Cosine, and Area
In your GCSEs, you learned how to handle right-angled triangles. At AS Level, we deal with any triangle. We usually label the angles with capital letters \(A, B, C\) and the sides opposite them with lowercase letters \(a, b, c\).
The Sine Rule
Use this when you have "matching pairs" (an angle and its opposite side).
Formula: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
The Cosine Rule
Use this when you have a "Side-Angle-Side" sandwich or when you know all three sides.
Finding a side: \(a^2 = b^2 + c^2 - 2bc \cos A\)
Finding an angle: \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)
Area of a Triangle
Forget "half base times height" for a moment. If you know two sides and the angle between them, use:
Formula: \(Area = \frac{1}{2}ab \sin C\)
Quick Review:
• Sine Rule: Best for pairs.
• Cosine Rule: Best for "sandwiches" (Side-Angle-Side).
• Check your calculator: Make sure you are in Degrees (D) mode for this chapter!
The "Ambiguous Case" of the Sine Rule:
Sometimes, when you use the Sine Rule to find an angle, there might be two possible triangles. This happens if you are given two sides and a non-enclosed acute angle.
Memory Trick: If the side opposite the angle is shorter than the other side, there might be two answers. One is \(\theta\), and the other is \(180^\circ - \theta\).
Takeaway: Always check if \(180^\circ - \theta\) could also fit in your triangle!
2. The Unit Circle and "CAST"
Have you ever wondered why \(\sin(150^\circ)\) is the same as \(\sin(30^\circ)\)? It’s all about the Unit Circle. Imagine a circle with a radius of 1. Any point on the edge of this circle has coordinates \(( \cos \theta, \sin \theta )\).
The CAST Diagram
This is a simple map to tell you which trig functions are positive in each quadrant:
• Quadrant 1 (0-90°): All are positive.
• Quadrant 2 (90-180°): Sine is positive.
• Quadrant 3 (180-270°): Tangent is positive.
• Quadrant 4 (270-360°): Cosine is positive.
Mnemonic: Add Sugar To Coffee (or All Students Take Calculus).
Did you know?
The word "sine" comes from the Latin word sinus, meaning "bay" or "curve." This refers to the way the side of the triangle looks when drawn inside a circle!
3. Trigonometric Graphs
You need to be able to recognize and sketch the three main graphs. They are periodic, meaning they repeat forever.
1. \(y = \sin x\): Starts at 0, goes up to 1 at \(90^\circ\), back to 0 at \(180^\circ\), down to -1 at \(270^\circ\), and back to 0 at \(360^\circ\). It looks like a smooth wave.
2. \(y = \cos x\): Starts at 1, goes to 0 at \(90^\circ\), down to -1 at \(180^\circ\), back to 0 at \(270^\circ\), and back to 1 at \(360^\circ\). It looks like a "bucket" or a shifted sine wave.
3. \(y = \tan x\): This one is different! It has asymptotes (lines it never touches) at \(90^\circ, 270^\circ\), etc. It repeats every \(180^\circ\).
Graph Transformations
Just like with other functions, you can move these graphs around:
• \(y = \sin(x + 30^\circ)\): Moves the graph left by \(30^\circ\).
• \(y = 2\cos x\): Stretches the graph vertically (now goes from 2 to -2).
• \(y = \tan(2x)\): Squeezes the graph horizontally (the period becomes \(90^\circ\) instead of \(180^\circ\)).
Takeaway: When sketching, always mark your intercepts (where it crosses the axes) and your maximum/minimum points clearly.
4. Trigonometric Identities
In algebra, we use identities to simplify expressions. In Trigonometry, you have two "best friends" that will help you solve almost any equation.
Identity 1: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
Use this if you see \(\tan\) and want to turn it into \(\sin\) and \(\cos\), or vice versa.
Identity 2: \(\sin^2 \theta + \cos^2 \theta = 1\)
This is just Pythagoras' Theorem in disguise! Use it to swap between \(\sin^2 \theta\) and \(\cos^2 \theta\).
Common Mistake to Avoid:
Remember that \(\sin^2 \theta\) means \((\sin \theta)^2\). It does not mean \(\sin(\theta^2)\)!
5. Solving Trig Equations
This is where everything comes together. You will often be asked to solve an equation like \(2\cos x = 1\) within a specific range, like \(0 \le x \le 360^\circ\).
Step-by-Step Guide:
Step 1: Isolate the trig function.
Example: \(2\cos x = 1 \rightarrow \cos x = 0.5\)
Step 2: Find the Principal Value (PV).
Use your calculator: \(x = \cos^{-1}(0.5) = 60^\circ\)
Step 3: Find other values in the range.
Use the symmetry of the graph or the CAST diagram.
• For Cosine, the other value is usually \(360^\circ - PV\). So, \(360^\circ - 60^\circ = 300^\circ\).
• For Sine, the other value is \(180^\circ - PV\).
• For Tangent, the other value is \(180^\circ + PV\).
Quadratic Trig Equations
Sometimes you’ll see something like \(2\sin^2 x - \sin x - 1 = 0\).
Don't panic! Just treat \(\sin x\) like a normal variable (like \(y\)).
Let \(y = \sin x\), so the equation becomes \(2y^2 - y - 1 = 0\).
Factorise it, find the values for \(y\), then solve for \(x\) at the end.
Takeaway: Always check your final answers are within the range given in the question (e.g., \(0\) to \(360^\circ\)).
Summary of Key Points
• Use Sine Rule for matching pairs and Cosine Rule for side-angle-side sandwiches.
• Use CAST or Graphs to find multiple solutions to equations.
• Memorize the two identities: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sin^2 \theta + \cos^2 \theta = 1\).
• Always check your calculator mode (Degrees!) and the interval asked for in the question.
Trigonometry can feel like a lot of rules at first, but with a bit of practice, you'll start to see the patterns. Keep at it!