Mastering Non-Right-Angled Triangles: Sine, Cosine, and Area Formulae
Welcome to one of the most practical chapters in Further Pure Mathematics! Up until now, you might have spent a lot of time working with right-angled triangles using SOH CAH TOA. But what happens when the triangle doesn't have a \(90^{\circ}\) angle? This chapter gives you the "master keys" to unlock the side lengths, angles, and areas of any triangle, no matter its shape.
Don't worry if you find trigonometry a bit intimidating at first. We will break these formulae down into simple patterns so you know exactly which tool to grab from your mathematical toolbox.
1. The Secret Language of Triangles: Labeling
Before using any formula, you must label your triangle correctly. If you get this wrong, the formulae won't work!
- We label the angles with capital letters: \(A\), \(B\), and \(C\).
- We label the sides with lowercase letters: \(a\), \(b\), and \(c\).
- The Golden Rule: Side \(a\) must be opposite angle \(A\). Side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\).
2. The Sine Rule
The Sine Rule is used when you have "matching pairs" of an angle and its opposite side.
The Formula:
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
When to use it:
1. When you know two angles and one side (to find a missing side).
2. When you know two sides and one non-included angle (to find a missing angle).
Pro Tip: If you are looking for a side, use the formula as written above. If you are looking for an angle, it is easier to flip the whole thing upside down:
\( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)
Quick Review: Remember that you only ever use two parts of the fraction at a time (e.g., \( \frac{a}{\sin A} = \frac{b}{\sin B} \)).
Common Pitfall: The Ambiguous Case
Sometimes, when using the Sine Rule to find an angle, there might be two possible answers (an acute angle and an obtuse angle). This happens because \( \sin \theta = \sin(180^{\circ} - \theta) \). Always check the diagram or the context of the question to see if the angle should be wider than \(90^{\circ}\).
3. The Cosine Rule
The Cosine Rule is your "heavy-duty" tool. It is slightly more complex, but it is provided on your formula sheet in the exam!
The Formula (for a side):
\( a^2 = b^2 + c^2 - 2bc \cos A \)
When to use it:
1. SAS (Side-Angle-Side): You know two sides and the angle between them.
2. SSS (Side-Side-Side): You know all three sides and want to find an angle.
Finding an Angle:
If you need to find an angle, you can rearrange the formula to:
\( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)
Did you know? The Cosine Rule is actually a more general version of Pythagoras' Theorem! If angle \(A\) is \(90^{\circ}\), then \( \cos 90^{\circ} = 0 \), and the formula turns into \( a^2 = b^2 + c^2 \).
4. Area of a Triangle
In younger years, you learned that Area \( = \frac{1}{2} \times \text{base} \times \text{height} \). However, finding the vertical height can be difficult. In Further Pure Maths, we use trigonometry instead.
The Formula:
\( \text{Area} = \frac{1}{2} ab \sin C \)
The Rule of Thumb: To use this formula, you need two sides and the included angle (the angle trapped between those two sides). If you have sides \(x\) and \(y\), you must use the angle where those two sides meet.
Example: If you have side \(a = 5\text{cm}\), side \(b = 7\text{cm}\), and angle \(C = 40^{\circ}\):
\( \text{Area} = \frac{1}{2} \times 5 \times 7 \times \sin 40^{\circ} \approx 11.25\text{cm}^2 \)
5. How to Choose the Right Method
Stuck on a problem? Follow this simple flowchart:
- Do I have a right-angled triangle?
Yes \(\implies\) Use SOH CAH TOA or Pythagoras.
No \(\implies\) Keep going. - Do I have a matching pair (Side \(a\) and Angle \(A\))?
Yes \(\implies\) Use the Sine Rule. - Do I have two sides and the angle between them?
Yes \(\implies\) Use the Cosine Rule (to find a side) or Area Formula. - Do I have all three sides?
Yes \(\implies\) Use the Cosine Rule (rearranged for the angle).
6. Key Takeaways and Exam Tips
1. Calculator Mode: Always check if your calculator is in Degrees (D) or Radians (R). Most triangle geometry questions in this section use degrees, but Section 10A (Radian Measure) will use radians. Read the question carefully!
2. Exact Values: The syllabus expects you to know the exact values for \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\). For example, \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \). If a question asks for an exact answer, don't use decimals!
3. Don't Round Too Early: Keep the full number on your calculator display during intermediate steps. Rounding your \( \cos A \) value to 1 decimal place halfway through a calculation can make your final answer very inaccurate.
4. Formula Sheet: Remember, the Cosine Rule is on the formula sheet provided in the exam, but the Sine Rule and Area Formula must be memorized!
Summary Table:
Tool: Sine Rule | Requirement: Matching side/angle pair.
Tool: Cosine Rule | Requirement: SAS or SSS.
Tool: Area | Requirement: Two sides and the included angle.