Non-Right-Angled Triangles: The Rules for ANY Triangle!
Welcome to the exciting world of trigonometry beyond right-angled triangles! Previously, you mastered SOH CAH TOA, but that brilliant tool only works if you have a 90° angle. What happens when you encounter an oblique triangle (a triangle with no right angle)?
That's where the Sine Rule, the Cosine Rule, and the advanced Area Formula come in. These powerful rules allow you to find unknown sides and angles in any triangle, making trigonometry truly universal.
Don't worry if these formulas look complex at first—they are all provided on your formula sheet in the exam! Your job is to understand when to use each one and how to apply them correctly.
1. Labelling a Non-Right-Angled Triangle
Before we use any rules, we need a consistent way to label our triangles. This is essential for plugging values into the formulas correctly.
- The three vertices (corners) are labelled with capital letters: A, B, C (representing the angles).
- The side opposite Angle A is labelled a (lowercase).
- The side opposite Angle B is labelled b (lowercase).
- The side opposite Angle C is labelled c (lowercase).
Key Tip: Always remember the side and its opposite angle share the same letter!
2. Area of a Triangle (The SAS Formula)
You already know the basic area formula: \(Area = \frac{1}{2} \times base \times height\). But what if the height is unknown?
We use the trigonometric area formula when we know two sides and the angle included (sandwiched) between them (SAS).
The Formula:
\[ Area = \frac{1}{2}ab \sin C \]
(Or, depending on the labels used: \(Area = \frac{1}{2}bc \sin A\) or \(Area = \frac{1}{2}ac \sin B\))
How to Use It (Step-by-Step):
- Identify the two sides you know (e.g., a and b).
- Identify the angle included between those two sides (e.g., Angle C).
- Substitute the values into the formula and calculate.
Analogy: Think of the angle as the filling in a sandwich. You need to know both slices of bread (the sides) and the filling (the included angle) to use this formula.
Quick Review: Area
- When to use: When you know SAS (Side, Angle, Side).
- Formula focus: \(\frac{1}{2} \times \text{side}_1 \times \text{side}_2 \times \sin(\text{Included Angle})\).
3. The Sine Rule (The "Pair" Rule)
The Sine Rule is used when you have enough information to form a complete pair: a known side and its opposite angle. It helps you find missing angles or sides when you have AAS (Angle, Angle, Side) or SSA (Side, Side, Angle).
The Formula:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Analogy: The Sine Rule works by creating a proportion. If you know the ratio of a side length to the sine of its opposite angle (a complete pair), you can find any other part of the triangle.
A. Finding a Missing Side (Put Sides on Top)
To find side \(a\), you use the relationship:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
(You must know \(b\), \(A\), and \(B\)).
Step 1: Set up the fraction with the unknown side on the top left:
\[ \frac{\text{unknown side}}{\sin (\text{opposite angle})} = \frac{\text{known side}}{\sin (\text{opposite angle})} \]
Step 2: Multiply both sides by the sine of the angle opposite the unknown side.
\[ a = \frac{b \sin A}{\sin B} \]
B. Finding a Missing Angle (Put Angles on Top)
When solving for an angle, it’s easier to flip the entire formula so the sines are on top. To find angle \(A\):
\[ \frac{\sin A}{a} = \frac{\sin B}{b} \]
Step 1: Set up the fraction with the unknown angle on the top left:
\[ \frac{\sin (\text{unknown angle})}{\text{opposite side}} = \frac{\sin (\text{known angle})}{\text{opposite side}} \]
Step 2: Multiply both sides by the side opposite the unknown angle, and then use the inverse sine function (\(\sin^{-1}\)) to find the angle.
Important Note: This process leads directly to the Ambiguous Case (See Section 5).
Quick Review: Sine Rule
- When to use: When you have a complete side-opposite-angle pair and one other piece of information (AAS or SSA).
- To find a side: Put sides on top.
- To find an angle: Put sines of angles on top.
4. The Cosine Rule (For SAS and SSS)
The Cosine Rule is your go-to tool when the Sine Rule can't be used—meaning you don't have a complete side-opposite-angle pair.
A. Finding a Missing Side (SAS Scenario)
This is used when you know two sides and the angle included between them, and you want to find the third side (SAS).
The Formula (Finding Side \(a\)):
\[ a^2 = b^2 + c^2 - 2bc \cos A \]
Analogy: Notice the first part, \(b^2 + c^2\), looks like Pythagoras! The second part, \(- 2bc \cos A\), is the correction factor needed because the triangle isn't right-angled.
Step-by-Step for Side \(a\):
- Identify the unknown side (\(a\)).
- The other two known sides are \(b\) and \(c\).
- The angle opposite the unknown side is \(A\).
- Substitute the values.
- Find the square root of the final result to get the side length \(a\).
B. Finding a Missing Angle (SSS Scenario)
If you know all three sides (SSS), you can rearrange the formula above to solve for any angle.
The Rearranged Formula (Finding Angle \(A\)):
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Step-by-Step for Angle \(A\):
- Identify the side opposite the angle you want to find (\(a\)). This side is the one being subtracted at the end.
- The other two sides are \(b\) and \(c\).
- Substitute the values.
- Use the inverse cosine function (\(\cos^{-1}\)) to find the angle \(A\).
Did you know? Unlike the Sine Rule, the Cosine Rule gives you the correct angle directly, whether it is acute or obtuse, because the cosine of an obtuse angle is negative, which is handled correctly by the calculator.
Quick Review: Cosine Rule
- When to use: When you have no complete side-opposite-angle pair.
- Use 1: Finding a side (requires SAS).
- Use 2: Finding an angle (requires SSS).
5. The Ambiguous Case of the Sine Rule (Extended Content)
This is a challenging but crucial topic for Extended students! The ambiguous case occurs only when you are given SSA (Side, Side, Angle) and you are using the Sine Rule to find a missing angle.
Why is it Ambiguous?
In geometry, there might be two possible triangles that fit the given SSA measurements. This is because the sine function is symmetrical between 0° and 180°:
\[ \sin(\theta) = \sin(180^\circ - \theta) \]
If your calculator gives you an acute angle (say, \(30^\circ\)), the angle could also be its obtuse counterpart (\(180^\circ - 30^\circ = 150^\circ\)).
When Must You Check for Ambiguity?
You must check for the ambiguous case every time you use the Sine Rule to find a missing angle.
The Rule for Checking:
After finding the first angle \(\theta_1\) using \(\sin^{-1}\):
- Calculate the alternative obtuse angle: \(\theta_2 = 180^\circ - \theta_1\).
- Check if this obtuse angle \(\theta_2\) can fit inside the triangle alongside the other known angle.
- If \((\text{known angle} + \theta_2) < 180^\circ\), then two valid triangles exist. You must state both possible values for the angle.
- If \((\text{known angle} + \theta_2) \ge 180^\circ\), then only the first (acute) triangle exists.
Example: You are given Angle A = 40°, side a = 8 cm, and side b = 11 cm. You use the Sine Rule to find Angle B, and your calculator gives \(\angle B_1 = 64.1^\circ\).
- Acute possibility: \(\angle B_1 = 64.1^\circ\). Check sum: \(40^\circ + 64.1^\circ = 104.1^\circ\). (Valid)
- Obtuse possibility: \(\angle B_2 = 180^\circ - 64.1^\circ = 115.9^\circ\). Check sum: \(40^\circ + 115.9^\circ = 155.9^\circ\). Since \(155.9^\circ < 180^\circ\), a second valid triangle exists.
Common Mistake to Avoid: Assuming the angle you calculate is the only answer. Always check the obtuse possibility when using Sine Rule to find an angle!
6. Which Rule to Choose? (The Decision Flowchart)
Choosing the correct formula is the first critical step in solving a triangle problem.
Decision Guide:
1. Do you know two sides and the angle INCLUDED between them (SAS)?
- To find the Area: Use \(Area = \frac{1}{2}ab \sin C\).
- To find the Opposite Side: Use the Cosine Rule (\(a^2 = b^2 + c^2 - 2bc \cos A\)).
2. Do you know all three sides (SSS)?
- To find any Angle: Use the Cosine Rule (rearranged form: \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)).
3. Do you have a complete side-opposite-angle pair? (i.e., you know \(a\) and \(A\))
- To find a Side: Use the Sine Rule (\(\frac{a}{\sin A} = \frac{b}{\sin B}\)).
- To find an Angle: Use the Sine Rule (\(\frac{\sin A}{a} = \frac{\sin B}{b}\)) — Remember to check for the Ambiguous Case!
Study Summary: Non-Right-Angled Triangles
Mastering these three formulas equips you to handle almost any 2D geometry problem.
Key Takeaways:
- Area Formula: Requires SAS (two sides and the included angle).
- Sine Rule: Used when you have a complete pair. Be extremely careful when finding angles due to the Ambiguous Case (check \(180^\circ - \theta\)).
- Cosine Rule: Used when you have SSS or SAS, but no complete pairs. This rule is safer for finding angles as it handles obtuse angles naturally.
Keep practising to make the decision on which rule to use quick and instinctive! You've got this!