Welcome to Trigonometry!
Trigonometry often gets a reputation for being one of the "tougher" chapters in A Level Maths, but it is actually one of the most useful and logical tools you will ever learn. At its heart, trigonometry is just about the relationship between the angles and sides of triangles. Whether you are interested in architecture, video game design, or understanding how sound waves work, trigonometry is the language you need.
In these notes, we will break down the Pearson Edexcel (9MA0) syllabus into bite-sized pieces. Don't worry if it seems tricky at first—we will take it one step at a time!
1. Radians and Circle Geometry
Until now, you have probably used degrees to measure angles. In A Level, we introduce radians. Think of radians as the "natural" language of circles. While 360 degrees is a bit of an arbitrary number, radians are based on the radius of the circle.
What is a Radian?
One radian is the angle formed when the arc length is exactly equal to the radius of the circle.
Key conversion: \(180^\circ = \pi \text{ radians}\)
Useful Formulae for Radians
When working in radians, circle calculations become much simpler:
1. Arc Length: \(s = r\theta\)
2. Area of a Sector: \(A = \frac{1}{2}r^2\theta\)
(Note: \(\theta\) must always be in radians for these to work!)
Quick Review Box:
- To go from Degrees to Radians: Multiply by \(\frac{\pi}{180}\)
- To go from Radians to Degrees: Multiply by \(\frac{180}{\pi}\)
Key Takeaway: Radians make circle math cleaner. Always check if your calculator is in "RAD" mode before starting a problem!
2. The Graphs and Exact Values
You need to be very familiar with the "waves" of \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\). They are periodic, meaning they repeat the same pattern forever.
The Unit Circle
Imagine a circle with a radius of 1. If you pick a point on the edge at an angle \(\theta\):
- The x-coordinate is \(\cos \theta\)
- The y-coordinate is \(\sin \theta\)
This is why \(\sin\) and \(\cos\) never go above 1 or below -1!
Exact Values You Must Know
You are expected to know the exact values for angles like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}\) and \(\frac{\pi}{2}\).
Example: \(\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\) and \(\cos(\frac{\pi}{3}) = \frac{1}{2}\).
Did you know? You can use your hand to remember these! If you label your fingers \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\), there are simple tricks to find the values instantly.
Key Takeaway: Learn the shapes of the graphs and the exact values. They are the building blocks for everything else.
3. Small Angle Approximations
When an angle \(\theta\) is very, very small (close to 0), the trig functions start to behave like simple linear or quadratic lines. This makes complex equations much easier to solve.
The Approximations (for \(\theta\) in radians):
1. \(\sin \theta \approx \theta\)
2. \(\tan \theta \approx \theta\)
3. \(\cos \theta \approx 1 - \frac{\theta^2}{2}\)
Common Mistake: Forgetting that these only work when \(\theta\) is in radians. If the question mentions degrees, convert first!
4. Reciprocal and Inverse Functions
This is where we meet the "cousins" of sine, cosine, and tangent.
Reciprocal Functions
These are just 1 divided by the original functions:
- Secant (sec): \(\sec \theta = \frac{1}{\cos \theta}\)
- Cosecant (cosec): \(\text{cosec } \theta = \frac{1}{\sin \theta}\)
- Cotangent (cot): \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\)
Memory Aid: Look at the third letter of the new function:
- sec \(\rightarrow\) cosine
- cosec \(\rightarrow\) sine
- cot \(\rightarrow\) tangent
Inverse Functions
Arcsin, Arccos, and Arctan are the inverse functions. We use these to find the angle when we already know the ratio.
Note: These have specific domains and ranges because they are only functions over a certain "snapshot" of the graph.
Key Takeaway: Reciprocals are \(1/f(x)\). Inverses are the "un-doing" of the function to find the angle.
5. Trigonometric Identities
Identities are equations that are always true. You use them to swap one expression for another to make a problem solvable.
The Fundamental Identities
1. \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
2. \(\sin^2 \theta + \cos^2 \theta = 1\)
New A Level Identities
By dividing \(\sin^2 \theta + \cos^2 \theta = 1\) by \(\cos^2 \theta\) or \(\sin^2 \theta\), we get:
- \(1 + \tan^2 \theta = \sec^2 \theta\)
- \(1 + \cot^2 \theta = \text{cosec}^2 \theta\)
Step-by-Step for Proofs:
1. Start with the more complicated side.
2. Change everything into \(\sin\) and \(\cos\).
3. Use identities to simplify.
4. Keep an eye on the "target" (the other side of the equation).
6. Addition and Double Angle Formulae
Sometimes we have angles added together, like \(\sin(A+B)\). You cannot just expand this like a normal bracket! You must use the specific formulae.
Addition (Compound Angle) Formulae
- \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
- \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)
(Watch out! The sign flips for cosine!)
Double Angle Formulae
These are just the addition formulae where \(A\) and \(B\) are the same:
- \(\sin 2A = 2 \sin A \cos A\)
- \(\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A\)
- \(\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}\)
Quick Review Box: The \(\cos 2A\) formula has three versions. Pick the one that helps you cancel out terms in your specific problem!
7. The \(R \cos(\theta \pm \alpha)\) Form
Sometimes you are given an expression like \(3 \sin \theta + 4 \cos \theta\) and asked to solve it. This is hard because there are two different trig terms. We can squash them into one single wave:
\(a \sin \theta + b \cos \theta = R \sin(\theta + \alpha)\)
How to find R and \(\alpha\):
1. R: \(R = \sqrt{a^2 + b^2}\) (It's just Pythagoras!)
2. \(\alpha\): Use \(\tan \alpha = \frac{\text{opposite coefficient}}{\text{adjacent coefficient}}\) (usually \(\frac{b}{a}\) or \(\frac{a}{b}\) depending on the form).
Analogy: Imagine two different waves clashing in the ocean. The \(R\) form tells you what the one "combined" wave looks like.
8. Solving Equations
This is where everything comes together. You will often be asked to solve for \(x\) in a given interval (e.g., \(0 \leq x \leq 360^\circ\)).
General Strategy:
1. Simplify: Use identities to get the equation into one type of trig function (e.g., all \(\sin\)).
2. Solve for the angle: Use your calculator's inverse function (\(\sin^{-1}\), etc.). This gives you the Principal Value.
3. Find other values: Use the symmetry of the graphs or a CAST diagram to find all other angles in the range.
4. Adjust: If the question was \(\sin(2x)\), solve for \(2x\) first, then divide your final answers by 2.
Key Takeaway: Always check your range. If the question is in \(x\), but you have \(2x\), you need to look for angles in a range twice as big!
9. Modelling with Trigonometry
Trig functions are perfect for modelling things that go up and down over time (periodic motion).
Examples:
- Tides: The depth of water in a harbor follows a \(\cos\) curve.
- Ferris Wheels: Your height above the ground as you spin.
- Daylight: The number of hours of sunlight throughout the year.
When solving these, "initial" usually means time \(t = 0\). Max and min values happen at the peaks and troughs of the waves.
Final Encouragement: Trigonometry is a skill that improves with practice. If a proof doesn't work the first time, try a different identity. You've got this!