Welcome to Non-Right-Angled Triangles!
Hi there! If you’ve mastered SOH CAH TOA and Pythagoras, that’s fantastic! But what happens when you need to calculate a distance or an angle in a triangle that doesn't have a \(90^\circ\) corner?
This chapter is your toolkit for tackling any triangle, no matter how strangely shaped it is. We will learn three powerful new formulas: the Area Formula, the Sine Rule, and the Cosine Rule. Don't worry, these formulas are provided on the formula list in the exam, but knowing how and when to use them is key!
1. Standard Notation for Triangles
Before we dive into the rules, we must agree on how to label a triangle. This notation is essential for applying the formulas correctly.
Rule:
- The angles are represented by capital letters (A, B, C).
- The sides are represented by the corresponding lowercase letters (a, b, c).
- A side is always opposite its corresponding angle.
Think of it like this: Side 'a' is always across the triangle from Angle 'A'.
2. Finding the Area of a Triangle
The New Area Formula: Side-Angle-Side (SAS)
You already know the basic area formula: Area \( = \frac{1}{2} \times \text{base} \times \text{height} \). But finding the perpendicular height in a non-right-angled triangle is often difficult.
The new formula lets us find the area if we know two sides and the angle between them (this is called the Included Angle).
The Formula
If you know sides \(a\) and \(b\), and the included angle \(C\):
Area \( = \frac{1}{2}ab \sin C \)
Note: You can swap the letters around. For example, if you know sides \(b\) and \(c\) and the angle \(A\), the formula becomes: Area \( = \frac{1}{2}bc \sin A \).
Memory Trick: The angle used must always be the one 'sandwiched' between the two sides you use in the formula!
3. The Sine Rule
The Sine Rule is used when you have a matching pair: a known side and its opposite angle. It helps you find missing sides or missing angles.
When to Use the Sine Rule
Use the Sine Rule when you are given:
- Two angles and one side (AAS or ASA).
- Two sides and a non-included angle (SSA - be careful! See the Ambiguous Case below).
The Formula (Used to Find Sides)
To find a missing side (a, b, or c), put the sides on top:
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
The Formula (Used to Find Angles)
To find a missing angle (A, B, or C), put the angles on top:
\( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)
Step-by-Step: Using the Sine Rule
- Identify the known pair (side and opposite angle). Set this equal to the ratio involving the unknown you want to find.
- Substitute your known values into the equation.
- Use cross-multiplication or rearranging to solve for the unknown side or \(\sin(\text{Angle})\).
- If finding an angle, remember to use the inverse sine function (\(\sin^{-1}\)) to find the final angle.
❗ Crucial Concept: The Ambiguous Case (SSA) ❗
This is the trickiest part of the Sine Rule, specifically when you are given Two Sides and a Non-Included Angle (SSA).
Because \(\sin(x^\circ) = \sin(180^\circ - x^\circ)\), there might be two possible triangles that fit the given information if the angle you are looking for is obtuse.
How to Handle It:
When solving for an unknown angle \(A\):
- Calculate the acute angle \(A_1 = \sin^{-1}(\dots)\).
- Check if a second, obtuse angle is possible: \(A_2 = 180^\circ - A_1\).
- You must check if the remaining angle in the triangle (\(180^\circ - A_2 - \text{given angle}\)) is still positive. If it is, then two solutions exist.
Example: If you find an angle \(A_1 = 30^\circ\). The second possible angle is \(180^\circ - 30^\circ = 150^\circ\). If the initial given angle was \(20^\circ\), both triangles (with \(30^\circ\) and \(150^\circ\)) are possible.
Key Takeaway for Sine Rule: You need a matching pair to start. Always consider the obtuse angle possibility when finding an unknown angle.
4. The Cosine Rule
The Cosine Rule is your go-to when you have sides and angles arranged in a way that doesn't provide a matching pair for the Sine Rule.
When to Use the Cosine Rule
Use the Cosine Rule when you are given:
- Two sides and the included angle (SAS) - used to find the third side.
- Three sides (SSS) - used to find any angle.
The Formula (Used to Find a Missing Side)
If you want to find side \(a\), and you know sides \(b\) and \(c\) and the included angle \(A\):
\( a^2 = b^2 + c^2 - 2bc \cos A \)
Analogy: Notice how the formula starts like Pythagoras' theorem (\(a^2 = b^2 + c^2\)) and then adds a correction factor (\(- 2bc \cos A\)). This correction handles the fact that it's not a right angle!
The Formula (Used to Find a Missing Angle)
We can rearrange the side formula to find an angle. If you want to find angle \(A\):
\( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)
Important: The side \(a^2\) being subtracted must be the side opposite the angle \(A\) you are finding. The sides \(b\) and \(c\) are the sides adjacent (next to) the angle \(A\).
Step-by-Step: Using the Cosine Rule
To Find a Side (SAS):
- Identify the side you are finding (e.g., \(a\)).
- Substitute the other two sides (\(b\) and \(c\)) and the included angle (\(A\)) into the formula \(a^2 = b^2 + c^2 - 2bc \cos A\).
- Calculate \(a^2\), then take the square root to find \(a\).
To Find an Angle (SSS):
- Identify the angle you want to find (e.g., \(A\)).
- Substitute all three sides into the rearranged formula: \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \). (Remember the opposite side is subtracted!).
- Calculate the value of \(\cos A\).
- Use the inverse cosine function (\(\cos^{-1}\)) to find the angle \(A\).
5. Deciding Which Rule to Use (The Trigonometry Checklist)
When you see a non-right-angled triangle problem, follow this flow chart:
Checklist for Finding Sides or Angles:
- Do I know two sides and the angle between them (SAS)? OR Do I know all three sides (SSS)?
If YES: Use the Cosine Rule. It is the safer option.
- Do I know a side and its opposite angle (a matching pair)?
If YES: Use the Sine Rule. It is usually faster, but watch out for the ambiguous case if solving for an angle (SSA).
- Is it a Right-Angled Triangle?
If YES: Use SOH CAH TOA or Pythagoras' Theorem.
⛔ Common Mistakes to Avoid ⛔
1. Mixing up the formulas: Always check if the side you are solving for (or the side you are subtracting in the angle formula) is opposite the angle you are using.
2. Forgetting to square root: When finding a side using the Cosine Rule, you calculate \(a^2\) first. Don't forget the final step of taking the square root!
3. Rounding too early: When using your answer from one step in the next step (especially in multi-part problems), use the full calculator value to maintain accuracy. Round only at the very end (typically to 3 significant figures or 1 decimal place for angles).
4. Ignoring the Ambiguous Case: If you use the Sine Rule to find an angle, and you are in the SSA scenario, always consider the possibility of \(180^\circ - \text{your angle}\).
✔ Chapter Key Takeaways ✔
1. Area Formula (SAS):
Area \( = \frac{1}{2}ab \sin C \). Uses two sides and the included angle.
2. Sine Rule:
Use for AAS, ASA, or SSA (caution!). Requires a known side/angle pair.
Side form: \( \frac{a}{\sin A} = \frac{b}{\sin B} \)
Angle form: \( \frac{\sin A}{a} = \frac{\sin B}{b} \)
Remember: Check for the Ambiguous Case \(180^\circ - A\) when finding an angle.
3. Cosine Rule:
Use for SSS or SAS.
Side form: \( a^2 = b^2 + c^2 - 2bc \cos A \)
Angle form: \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)
It handles obtuse angles automatically!
You now have the mathematical strength to conquer any 2D triangle problem thrown your way. Keep practicing which rule to choose, and you’ll find these problems become much simpler!