Welcome to the World of Trigonometry!
Welcome! Trigonometry might sound like a mouthful, but at its heart, it is simply the study of the relationships between the sides and angles of triangles. However, in A-Level Mathematics, we take it a step further. We look at periodic functions—things that repeat over and over, like waves, heartbeats, or even the phases of the moon. Whether you are aiming for an A* or just trying to get your head around the basics, these notes are here to guide you step-by-step.
1. The Fundamentals: Radians and Degrees
Before we dive into graphs, we need to know how to measure angles. While you are used to degrees (0° to 360°), mathematicians often use radians.
Quick Review:
A full circle is \(360^{\circ}\) or \(2\pi\) radians.
To convert Degrees to Radians: Multiply by \(\frac{\pi}{180}\).
To convert Radians to Degrees: Multiply by \(\frac{180}{\pi}\).
2. The "Big Three" Graphs
You need to be able to sketch and recognize the graphs of sine, cosine, and tangent. These graphs are "periodic," meaning they repeat their shape.
The Sine Graph \(y = \sin x\)
Starts at \((0,0)\), goes up to \(1\), down to \(-1\), and returns to the middle at \(360^{\circ}\) (or \(2\pi\)). It looks like a smooth wave.
The Cosine Graph \(y = \cos x\)
Starts at its peak \((0,1)\), goes down to \(-1\), and returns to \(1\) at \(360^{\circ}\). It is essentially the Sine wave shifted to the left.
The Tangent Graph \(y = \tan x\)
This one is different! It has asymptotes (lines it never touches) at \(90^{\circ}, 270^{\circ}\), etc. It goes from negative infinity to positive infinity.
Don't forget transformations: You might be asked to sketch \(y = 3\sin x\) (a vertical stretch) or \(y = \cos(2x)\) (a horizontal squish). Always label your axes clearly!
Takeaway: Sine and Cosine live between \(-1\) and \(1\). Tangent is the "wild child" that goes forever up and down.
3. Exact Values: The Hidden Shortcuts
In many exam questions, you won't use a calculator. You need to know the exact values for \(30^{\circ}, 45^{\circ},\) and \(60^{\circ}\).
Key Values to Memorize:
\(\sin 30^{\circ} = \frac{1}{2}\)
\(\sin 45^{\circ} = \frac{1}{\sqrt{2}}\)
\(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)
\(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
\(\cos 60^{\circ} = \frac{1}{2}\)
\(\tan 45^{\circ} = 1\)
Memory Tip: Notice how \(\sin 30^{\circ}\) is the same as \(\cos 60^{\circ}\)? They are "complementary" because \(30 + 60 = 90\).
4. The Core Identities (The "Bread and Butter")
Identities are equations that are true for every angle. These are your best friends for simplifying complicated expressions.
1. The Tangent Identity:
\(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\)
2. The Pythagorean Identity:
\(\sin^2 \theta + \cos^2 \theta \equiv 1\)
Common Mistake: Students often think \(\sin \theta^2\) is the same as \(\sin^2 \theta\). It’s not! \(\sin^2 \theta\) means \((\sin \theta) \times (\sin \theta)\).
5. Solving Equations: The CAST Diagram
When solving an equation like \(\sin \theta = 0.5\), your calculator only gives you one answer (\(30^{\circ}\)). But there are usually more! Use the CAST diagram or the graph to find them.
Step-by-Step Process:
1. Find the Basic Angle (the calculator answer, ignoring any minus signs).
2. Draw the CAST diagram to see which Quadrants the ratio is positive or negative in.
3. Calculate the other angles within the required range (e.g., \(0^{\circ}\) to \(360^{\circ}\)).
Memory Aid for CAST:
C - Cosine is positive here (4th Quadrant)
A - All are positive here (1st Quadrant)
S - Sine is positive here (2nd Quadrant)
T - Tangent is positive here (3rd Quadrant)
Mnemonic: "All Science Teachers are Crazy" or "Add Sugar To Coffee"
6. New Reciprocal Functions (Pure 2/3)
As you progress, you will meet three new functions. They are just the "flips" (reciprocals) of the ones you already know.
Secant (sec): \(\sec \theta = \frac{1}{\cos \theta}\)
Cosecant (csc or cosec): \(\csc \theta = \frac{1}{\sin \theta}\)
Cotangent (cot): \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\)
How to remember which is which? Look at the third letter of the new function!
sec \(\rightarrow\) cosine
cosec \(\rightarrow\) sine
cot \(\rightarrow\) tangent
7. Advanced Identities
For Pure 2 and 3, you need to master these to solve tougher equations.
Squared Identities:
\(1 + \tan^2 \theta \equiv \sec^2 \theta\)
\(1 + \cot^2 \theta \equiv \csc^2 \theta\)
Compound Angle Formulas:
These help you deal with angles being added or subtracted inside the function:
\(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
\(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\) (Careful! The sign flips for cosine!)
Double Angle Formulas:
Crucial for simplifying! They are derived from the compound formulas where \(A = B\).
\(\sin 2A = 2 \sin A \cos A\)
\(\cos 2A = \cos^2 A - \sin^2 A\)
(Also: \(\cos 2A = 2\cos^2 A - 1\) or \(1 - 2\sin^2 A\))
8. The R-Formula
Sometimes you are asked to combine a sine and cosine term into one, like \(3\sin \theta + 4\cos \theta\). We turn this into a single wave:
\(R \sin(\theta + \alpha)\) or \(R \cos(\theta - \alpha)\)
How to do it:
1. \(R = \sqrt{a^2 + b^2}\) (It's just Pythagoras!)
2. \(\alpha = \tan^{-1}(\frac{b}{a})\)
This is super useful for finding the maximum or minimum values of a function.
Final Tips for Success
Don't panic! Trigonometry is very visual. If you get stuck on an equation, try sketching the graph. It often makes the answer obvious.
Quick Review Box:
- Always check if your calculator is in Degrees or Radians mode before you start!
- \(\sin^2 \theta + \cos^2 \theta\) is always \(1\).
- If you see a "squared" trig term in an equation, look for a way to use the squared identities.
- Range matters! If the question asks for \(0 \leqslant x \leqslant \pi\), give your answers in radians.
Did you know? Trigonometry was originally used by ancient astronomers to map the stars and navigate the oceans. Today, it’s used to create the graphics in your favorite video games!