Welcome to Basic Trigonometry!
Welcome to one of the most useful chapters in A Level Maths! Trigonometry is all about the relationships between the sides and angles of triangles. While it might seem like a lot of formulas at first, it's actually just a set of tools to help you navigate shapes. Whether you're planning to be an architect, a pilot, or a game developer, trigonometry is your best friend.
Don't worry if you've forgotten some of your GCSE basics; we'll recap those first before diving into the exciting A Level content!
1. Right-Angled Triangles: The Foundation
Before we do anything else, we must be able to label a right-angled triangle correctly based on a specific angle, usually called \(\theta\) (theta).
- Hypotenuse: The longest side, opposite the right angle.
- Opposite: The side across from angle \(\theta\).
- Adjacent: The side next to angle \(\theta\) (that isn't the hypotenuse).
The SOH CAH TOA Mnemonic
This is the classic way to remember the three primary ratios:
- SOH: \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
- CAH: \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
- TOA: \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)
Angle Measurement: Degrees and Radians
In A Level Mathematics, angles can be measured in degrees (\(^\circ\)) or radians (\(\text{rad}\)), where \(\pi\text{ rad} = 180^\circ\). Always check your calculator mode before calculating: ensure it is in 'D'/'DEG' for degrees or 'R'/'RAD' for radians as required by the question.
Key Takeaway: If you have a right-angled triangle and know two pieces of information (like one side and one angle), you can find everything else using SOH CAH TOA.
2. The Sine and Cosine Rules
What if the triangle doesn't have a right angle? Don't panic! We have two powerful rules for "non-right-angled" triangles.
The Sine Rule
Use this when you have "matching pairs" (an angle and its opposite side):
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
Note: You can also flip these fractions upside down if you are trying to find an angle: \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\).
The Ambiguous Case of the Sine Rule
When given two sides and a non-included angle (SSA), there may be two possible triangles (one where the unknown angle is acute, and another where it is obtuse: \(180^\circ - \theta\) or \(\pi - \theta\)). Always check if an obtuse solution is geometrically possible (i.e. the sum of angles remains below \(180^\circ\)).
The Cosine Rule
Think of this as Pythagoras' Theorem's older, more sophisticated brother. Use it when you have two sides and the angle between them (the "SAS" case), or all three sides.
\(a^2 = b^2 + c^2 - 2bc \cos A\)
To find an angle, rearrange it to:
\(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)
Area of a Triangle
You no longer need the vertical height! If you know two sides (\(a\) and \(b\)) and the included angle (\(C\)):
\(\text{Area} = \frac{1}{2}ab \sin C\)
Did you know?
The Sine Rule is used extensively in Bearings and navigation. If a ship sails 10km on a bearing of \(045^\circ\), you can use these rules to find exactly how far North and East it has moved.
Key Takeaway: Use the Sine Rule for matching pairs and the Cosine Rule for "side-angle-side" or "side-side-side" setups.
3. Exact Trig Values
In OCR A Level Mathematics B (MEI), a calculator is permitted on all exam papers, but you are required to know and use exact trigonometric values to give exact analytical answers and show complete working without relying solely on decimal approximations.
\(0^\circ\) (\(0\) rad): \(\sin(0) = 0\), \(\cos(0) = 1\), \(\tan(0) = 0\)
\(30^\circ\) (\(\frac{\pi}{6}\) rad): \(\sin(30^\circ) = \frac{1}{2}\), \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), \(\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)
\(45^\circ\) (\(\frac{\pi}{4}\) rad): \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\), \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), \(\tan(45^\circ) = 1\)
\(60^\circ\) (\(\frac{\pi}{3}\) rad): \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), \(\cos(60^\circ) = \frac{1}{2}\), \(\tan(60^\circ) = \sqrt{3}\)
\(90^\circ\) (\(\frac{\pi}{2}\) rad): \(\sin(90^\circ) = 1\), \(\cos(90^\circ) = 0\), \(\tan(90^\circ)\) is undefined
Memory Aid: The Square Root Trick
To remember \(\sin \theta\) for \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\) (or \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\)), write down:
\(\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\)
This simplifies exactly to \(0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1\)!
Key Takeaway: Practice writing out the exact value table in both degrees and radians until you can do it from memory in under 30 seconds.
4. The Unit Circle and Any Angle
Until now, angles were often between \(0^\circ\) and \(90^\circ\). But in A Level, we look at angles like \(150^\circ\) (\(\frac{5\pi}{6}\)) or even negative angles and angles greater than \(360^\circ\). We use the Unit Circle (a circle of radius 1 centered at the origin) to define this.
- The x-coordinate on the circle is \(\cos \theta\).
- The y-coordinate on the circle is \(\sin \theta\).
- The gradient of the radius line is \(\tan \theta\).
The CAST Diagram
Because the circle repeats, different angles can have the same trig value. The CAST diagram tells you where the trigonometric ratios are positive:
- Quadrant 1 (\(0\) to \(90^\circ\) / \(0\) to \(\frac{\pi}{2}\)): All are positive.
- Quadrant 2 (\(90^\circ\) to \(180^\circ\) / \(\frac{\pi}{2}\) to \(\pi\)): Only Sin is positive.
- Quadrant 3 (\(180^\circ\) to \(270^\circ\) / \(\pi\) to \(\frac{3\pi}{2}\)): Only Tan is positive.
- Quadrant 4 (\(270^\circ\) to \(360^\circ\) / \(\frac{3\pi}{2}\) to \(2\pi\)): Only Cos is positive.
Key Takeaway: Use the symmetry of the unit circle or the CAST diagram to find "secondary" solutions for trig equations.
5. Trig Graphs
Visualizing the functions helps you understand their behavior.
- \(\sin \theta\): Starts at \((0,0)\), waves between \(1\) and \(-1\). Has a period of \(360^\circ\) (or \(2\pi\) radians).
- \(\cos \theta\): Starts at \((0,1)\), waves between \(1\) and \(-1\). It is the sine graph shifted left by \(90^\circ\) (\(\frac{\pi}{2}\)). Period is \(360^\circ\) (\(2\pi\)).
- \(\tan \theta\): Has a period of \(180^\circ\) (\(\pi\)) and vertical asymptotes at \(90^\circ, 270^\circ\) (\(\frac{\pi}{2}, \frac{3\pi}{2}\)), etc.
Analogy: The Ferris Wheel
Think of the \(\sin \theta\) graph like a Ferris wheel. As the wheel turns (angle \(\theta\)), your height above the ground goes up and down in a smooth, repeating wave. That is exactly what a sine wave represents!
Key Takeaway: Knowing the "shape" of the graphs helps you spot mistakes. For example, if you calculate \(\sin \theta = 2\), you know it's impossible because the graph never goes above 1!
6. Trigonometric Identities
Identities are equations that are always true for all valid angles. They are used to simplify complex expressions or solve equations.
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 Alert!
Note the notation: \(\sin^2 \theta\) means \((\sin \theta)^2\). It is not the same as \(\sin(\theta^2)\). Be very careful where you put that little '2'!
Key Takeaway: If an equation has both \(\sin \theta\) and \(\cos \theta\), try using an identity to change everything into just one type of trig function.
7. Solving Trigonometric Equations
This is where everything comes together. You'll often be asked to solve something like \(\sin(2\theta) = 0.5\) for \(0^\circ \le \theta \le 360^\circ\) (or \(0 \le \theta \le 2\pi\)).
Step-by-Step Guide:
- Isolate: Get the trig function by itself (e.g., \(\sin(2\theta) = 0.5\)).
- Find the Modified Range: If the argument is \(2\theta\) and \(0^\circ \le \theta \le 360^\circ\), adjust your working interval: \(0^\circ \le 2\theta \le 720^\circ\).
- Principal Value: Use the inverse trig function (e.g., \(2\theta = \sin^{-1}(0.5) = 30^\circ\)).
- Secondary Values & Periodic Solutions: Use the CAST diagram or graph symmetries (for sine: \(180^\circ - \text{PV}\); for cosine: \(360^\circ - \text{PV}\); for tan: \(\text{PV} + 180^\circ\)) and add/subtract periods (\(360^\circ\) / \(2\pi\)) to find all values of \(2\theta\) within the modified range.
- Final Adjustment: Divide each angle by 2 to find \(\theta\).
Don't worry if this seems tricky...
Solving equations with \(2\theta\) or \(3\theta\) is one of the most common places students lose marks. Just remember: Always find all the angles within the modified interval first, THEN divide at the very end.
Key Takeaway: Most trig equations have more than one answer. Always check your range and use your graph/CAST diagram to find them all!