Welcome to the World of Trigonometry!
Welcome! Trigonometry might seem like a scary word, but at its heart, it’s just the study of the relationships between the sides and angles of triangles. In A Level Mathematics (Paper 1), we go beyond the basic triangles you saw in GCSE and explore periodic functions—patterns that repeat forever, like sound waves or the tides. Whether you are aiming for an A* or just want to pass comfortably, these notes will break everything down into simple, manageable steps.
1. Radians: The Professional Way to Measure Angles
In GCSE, you used degrees. In A Level, we primarily use Radians. Think of radians as the "natural" language of circles. While 360 degrees is an arbitrary number, radians are based on the radius of the circle itself.
The Golden Rule: \(180^\circ = \pi \text{ radians}\).
To convert:
- Degrees to Radians: Multiply by \(\frac{\pi}{180}\)
- Radians to Degrees: Multiply by \(\frac{180}{\pi}\)
Arc Length and Sector Area
Because radians are based on the radius, the formulas for circles become much simpler:
- Arc Length (\(s\)): \(s = r\theta\)
- Area of a Sector (\(A\)): \(A = \frac{1}{2}r^2\theta\)
Note: For these formulas to work, \(\theta\) must be in radians!
Quick Review: Keep your calculator in RAD mode unless the question specifically uses degrees (\(^\circ\)). This is the most common mistake students make!
Key Takeaway
Radians make circular math easier. Remember that \(\pi\) is just a half-turn (\(180^\circ\)).
2. Trigonometric Graphs and Exact Values
You need to know the shapes of \(y = \sin x\), \(y = \cos x\), and \(y = \tan x\). They are periodic, meaning they repeat their shape every \(360^\circ\) (or \(2\pi\) radians). For \(\tan x\), it repeats every \(180^\circ\) (\(\pi\) radians).
The "CAST" Diagram (or the Unit Circle)
Trig functions can be positive or negative depending on the angle. A great way to remember this is the CAST diagram:
- Quadrant 1 (0 to \(\pi/2\)): All are positive.
- Quadrant 2 (\(\pi/2\) to \(\pi\)): Sine is positive.
- Quadrant 3 (\(\pi\) to \(3\pi/2\)): Tangent is positive.
- Quadrant 4 (\(3\pi/2\) to \(2\pi\)): Cosine is positive.
Mnemonic: All Students Take Calculus.
Exact Values You Must Know
Don't rely on your calculator for everything! You need to know values for \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \text{ and } \frac{\pi}{2}\).
Trick: For \(\sin \theta\), the values are \(\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\). Simplify these, and you have the exact values for \(0^\circ\) to \(90^\circ\)!
Key Takeaway
Symmetry is your friend. Use the CAST diagram or the graph's wave-shape to find multiple solutions to equations.
3. Reciprocal and Inverse Functions
This is where A Level adds more "ingredients" to the mix. We introduce three new "reciprocal" functions (flipping the fractions):
- Secant (sec): \(\sec x = \frac{1}{\cos x}\) (The 3rd letter is c, so it goes with cos)
- Cosecant (cosec): \(\csc x = \frac{1}{\sin x}\) (The 3rd letter is s, so it goes with sin)
- Cotangent (cot): \(\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\)
Inverse Functions (arcsin, arccos, arctan)
These are the "undo" buttons. If \(\sin(30^\circ) = 0.5\), then \(\arcsin(0.5) = 30^\circ\).
Important: To make these true functions, we restrict their range. For example, \(\arcsin x\) only gives answers between \(-\pi/2\) and \(\pi/2\).
Key Takeaway
Reciprocals (\(\sec, \csc, \cot\)) are not the same as inverses (\(\arcsin, \arccos, \arctan\)). Don't mix them up!
4. Trigonometric Identities
Identities are equations that are always true. They are the "tools" you use to simplify complex expressions or solve equations.
The Fundamentals:
- \(\tan \theta \equiv \frac{\sin \theta}{\cos \theta}\)
- \(\sin^2 \theta + \cos^2 \theta \equiv 1\)
The New A Level Identities (Derived from \(\sin^2 \theta + \cos^2 \theta \equiv 1\)):
- \(1 + \tan^2 \theta \equiv \sec^2 \theta\)
- \(1 + \cot^2 \theta \equiv \csc^2 \theta\)
Did you know? You don't have to memorize the last two! Just take \(\sin^2 \theta + \cos^2 \theta = 1\) and divide every term by \(\cos^2 \theta\) to get the first one, or by \(\sin^2 \theta\) to get the second one!
Key Takeaway
Whenever you see a \(\sin^2\) and a \(\cos^2\) in the same problem, look for a way to use the identity \(\sin^2 \theta + \cos^2 \theta = 1\).
5. Addition and Double Angle Formulae
These formulas allow you to break down angles like \((A + B)\) or \(2A\). These are provided in the formula booklet, but you must know how to use them!
Addition Formulae
\(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
\(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\) (Note: the sign flips for cosine!)
Double Angle Formulae
By letting \(B = A\) in the addition formulas, we get:
- \(\sin 2A = 2 \sin A \cos A\)
- \(\cos 2A = \cos^2 A - \sin^2 A\) (Which can also be written as \(2\cos^2 A - 1\) or \(1 - 2\sin^2 A\))
Step-by-Step Tip: When solving an equation like \(\cos 2x + \sin x = 0\), use the double angle formula to turn everything into \(\sin x\) so you can solve it like a quadratic equation.
Key Takeaway
Double angle formulas are just specific versions of addition formulas. Use them to make all the angles in your equation the same.
6. The Harmonic Form: \(R \cos(\theta \pm \alpha)\)
Sometimes you are given a mix like \(3 \cos \theta + 4 \sin \theta\). This is hard to solve. We can combine them into one single wave: \(R \cos(\theta - \alpha)\).
- \(R\) is the amplitude (how high the wave goes). \(R = \sqrt{a^2 + b^2}\).
- \(\alpha\) is the phase shift (how much the wave moved left or right). Found using \(\tan \alpha = \frac{b}{a}\).
Analogy: Combining two different sounds (\(\sin\) and \(\cos\)) to create one new, louder sound with its own unique timing.
Key Takeaway
Use this form to find the maximum or minimum values of a function quickly. The maximum of \(R \cos(\theta - \alpha)\) is just \(R\)!
7. Small Angle Approximations
When an angle \(\theta\) is very, very small (and measured in radians), the trig functions behave in very predictable ways:
- \(\sin \theta \approx \theta\)
- \(\tan \theta \approx \theta\)
- \(\cos \theta \approx 1 - \frac{\theta^2}{2}\)
Don't worry if this seems odd! It just means that for tiny angles, the curve of a sine wave looks almost exactly like a straight diagonal line.
Key Takeaway
This is a "substitution" topic. If the question says \(\theta\) is small, replace \(\sin \theta\) with \(\theta\) and simplify the algebra.
8. Solving Trigonometric Equations
This is where it all comes together. Here is your fail-proof checklist:
- Isolate the trig function (e.g., get it to \(\sin(2x) = 0.5\)).
- Find the Principal Value using your calculator (\(\sin^{-1}(0.5) = 30^\circ\)).
- Find Other Values in the range using the CAST diagram or the graph.
- Adjust for the angle. If the angle was \(2x\), you now divide all your answers by 2.
Common Mistake: Dividing by a trig function (e.g., dividing both sides by \(\cos x\)). This can "delete" potential solutions! Instead, factorise the equation.
Key Takeaway
Always check your range. If the question asks for solutions between \(0\) and \(2\pi\), don't give your answer in degrees!
Final Encouragement
Trigonometry is a "cumulative" topic. The more you practice using the identities, the more they will feel like second nature. Don't be afraid to sketch the graphs whenever you get stuck—they are the best map you have! You've got this!