Welcome to the World of Trigonometry!
Hi there! Welcome to one of the most useful and fascinating chapters in your Mathematics journey: Trigonometry. Whether you are aiming to be an engineer, an architect, or just want to pass your exams with flying colors, trigonometry is your toolkit for understanding shapes, waves, and movement.
Don't worry if you’ve found triangles a bit "pointy" or confusing in the past. We are going to break everything down step-by-step. By the end of these notes, you’ll be solving complex equations and sketching graphs like a pro!
1. Solving Any Triangle: Sine and Cosine Rules
In your earlier years, you learned about right-angled triangles. But what if the triangle doesn't have a \(90^\circ\) angle? That’s where the Sine Rule and Cosine Rule come to the rescue.
The Sine Rule
Think of the Sine Rule as a way to link opposite pairs. If you know a side and its opposite angle, you’re halfway there!
The Formula: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
Example: Use this when you have two angles and one side, or two sides and an angle that is NOT between them.
The Ambiguous Case (The "Double Trouble")
Sometimes, the Sine Rule can give you two possible triangles. This happens when you are given two sides and a non-included acute angle. One triangle might have an acute angle, and the other an obtuse angle (an angle bigger than \(90^\circ\)).
Quick Review: If \(\sin \theta = 0.5\), \(\theta\) could be \(30^\circ\) or \(180 - 30 = 150^\circ\). Always check if both fit in your triangle!
The Cosine Rule
Think of this as the "Power Version" of Pythagoras' Theorem. It works for any triangle.
The Formula: \(a^2 = b^2 + c^2 - 2bc \cos A\)
When to use it:
1. When you know two sides and the angle between them (SAS).
2. When you know all three sides and want to find an angle (SSS).
Area of a Triangle
Forget "half base times height" for a moment. If you have two sides and the angle between them, use this:
Area = \(\frac{1}{2}ab \sin C\)
Key Takeaway:
Use Sine Rule for pairs; use Cosine Rule for "side-angle-side" or "three sides".
2. Radians: A New Way to Measure
Up until now, you’ve used degrees. But in advanced math, we use Radians. A radian is just a different "language" for angles, based on the radius of a circle.
The Golden Rule: \(\pi \text{ radians} = 180^\circ\)
Converting is Easy:
- To go from Degrees to Radians: Multiply by \(\frac{\pi}{180}\).
- To go from Radians to Degrees: Multiply by \(\frac{180}{\pi}\).
Arc Length and Sector Area
When using radians, the formulas for circles become much simpler!
1. Arc Length (s): \(s = r\theta\)
2. Area of a Sector (A): \(A = \frac{1}{2}r^2\theta\)
Note: For these formulas to work, \(\theta\) MUST be in radians.
Common Mistake: Forgetting to change your calculator mode! If the question has \(\pi\) or says "rad," make sure your calculator shows a little 'R' at the top, not a 'D'.
3. Trigonometric Graphs and Transformations
Trig functions aren't just numbers; they are waves! You need to know the shapes of \(y = \sin x\), \(y = \cos x\), and \(y = \tan x\).
Key Properties:
- Sine and Cosine: They repeat every \(360^\circ\) (or \(2\pi\) radians). This is called their period. They wave between \(1\) and \(-1\).
- Tangent: This one is the "rebel." It repeats every \(180^\circ\) (or \(\pi\) radians) and has asymptotes (lines the graph never touches) at \(90^\circ, 270^\circ\), etc.
Transforming the Waves
Imagine the graph is a piece of string. You can stretch it or shift it:
- \(y = 3 \sin x\): The wave gets taller (reaches \(3\) and \(-3\)).
- \(y = \sin 2x\): The wave gets squashed horizontally (you fit two waves where there used to be one).
- \(y = \sin(x + 30^\circ)\): The whole wave shifts left by \(30^\circ\).
Did you know?
The sine wave is the shape of pure sound. When you hear a clear whistle or a tuning fork, you are literally hearing a \(\sin x\) graph!
4. Trigonometric Identities
Identities are mathematical facts that are always true. They are like shortcuts that help you simplify messy equations.
Identity 1: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
Identity 2: \(\sin^2 \theta + \cos^2 \theta = 1\)
Tip: You can rearrange Identity 2! For example, \(\sin^2 \theta = 1 - \cos^2 \theta\). This is super useful when you have an equation with both \(\sin\) and \(\cos\) in it.
5. Solving Equations
This is where it all comes together. You might be asked to find \(x\) for an equation like \(\sin(x + 30^\circ) = 0.5\).
Step-by-Step Guide:
- Isolate the trig function: Get it so it looks like \(\sin(\text{something}) = \text{number}\).
- Find the Principal Value: Use your calculator (\(\sin^{-1}\)). Let's call this angle \(PV\).
- Find other values in the range:
- For Sine: \(PV\) and \(180 - PV\)
- For Cosine: \(PV\) and \(360 - PV\) (or \(-PV\))
- For Tangent: \(PV\) and \(180 + PV\)
- Adjust for brackets: If the question was \(\sin(2x)\), you find the angles first, then divide them all by \(2\) at the very end.
Memory Aid: CAST Diagram
To remember where functions are positive:
Cosine is + in 4th quadrant.
All are + in 1st quadrant.
Sine is + in 2nd quadrant.
Tan is + in 3rd quadrant.
(Mnemonic: Castle Age Starts Today or All Stations To Central)
Quadratic Trig Equations
If you see something like \(6\cos^2 x + \sin x - 5 = 0\), don't panic! Use your identity to turn \(\cos^2 x\) into \(1 - \sin^2 x\). Now it's just a regular quadratic equation where "x" is \(\sin x\). Treat \(\sin x\) like a single letter (like \(y\)), solve the quadratic, then solve for \(x\).
Key Takeaway:
Always find the first angle using your calculator, then use the symmetries of the unit circle (or CAST) to find the rest within the requested interval.
Congratulations! You've just covered the core of AS Level Trigonometry. Keep practicing those sketches and keep an eye on your calculator modes. You've got this!