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: \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)
Quick Review: Calculator Checklist
Always check your calculator mode! For this chapter, make sure it says 'D' or 'DEG' for degrees. If you get a math error or a very strange decimal, this is usually the culprit.
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!
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 angle between them (\(C\)):
$$\text{Area} = \frac{1}{2}ab \sin C$$
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 your exam, you are expected to know the exact values for specific angles without using a calculator. This is a common "non-calculator style" question area.
30°: \(\sin(30) = \frac{1}{2}\), \(\cos(30) = \frac{\sqrt{3}}{2}\), \(\tan(30) = \frac{1}{\sqrt{3}}\)
45°: \(\sin(45) = \frac{\sqrt{2}}{2}\), \(\cos(45) = \frac{\sqrt{2}}{2}\), \(\tan(45) = 1\)
60°: \(\sin(60) = \frac{\sqrt{3}}{2}\), \(\cos(60) = \frac{1}{2}\), \(\tan(60) = \sqrt{3}\)
0° and 90°: \(\sin(0)=0\), \(\cos(0)=1\), \(\sin(90)=1\), \(\cos(90)=0\)
Memory Aid: The Square Root Trick
To remember \(\sin \theta\) for \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\), 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 the values above!
Key Takeaway: Practice writing out the exact value table until you can do it from memory in under 30 seconds.
4. The Unit Circle and Any Angle
Until now, angles were always between \(0^\circ\) and \(90^\circ\). But in A Level, we look at angles like \(150^\circ\) or even \(300^\circ\). We use the Unit Circle (a circle with radius 1) 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 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 values are positive:
- Quadrant 1 (0-90°): All are positive.
- Quadrant 2 (90-180°): Only Sin is positive.
- Quadrant 3 (180-270°): Only Tan is positive.
- Quadrant 4 (270-360°): 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. Repeats every \(360^\circ\).
- \(\cos \theta\): Starts at (0,1), waves between 1 and -1. It’s just the sine graph shifted left by \(90^\circ\).
- \(\tan \theta\): Has Asymptotes (lines it never touches) at \(90^\circ, 270^\circ\), etc. It goes off to infinity!
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. They are used to simplify complex expressions or solve equations.
1. The Tangent Identity: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
2. The Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\)
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 \(0^\circ \le \theta \le 360^\circ\).
Step-by-Step Guide:
- Isolate: Get the trig function by itself (e.g., \(\sin \theta = \dots\)).
- Principal Value: Use the inverse function on your calculator (e.g., \(\theta = \sin^{-1}(0.5)\)).
- Find Range: If the angle is \(2\theta\), adjust your search range (e.g., if \(0 \le \theta \le 360\), then \(0 \le 2\theta \le 720\)).
- Secondary Values: Use the CAST diagram or graph symmetry to find other angles in the range.
- Final Adjustment: If you were solving for \(2\theta\), divide your final answers by 2 to get \(\theta\).
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 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!