Understanding the Sine Rule
The sine rule relates the lengths of sides of a triangle to the sines of their opposite angles. It states that for any triangle ABC with sides a, b, and c opposite angles A, B, and C respectively:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
This rule is especially useful when you know either two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). It allows you to find unknown sides or angles by setting up proportions.
- Sine rule formula: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
- Useful for ASA, AAS, and SSA cases
- Requires knowledge of at least one side and its opposite angle
Applying the Cosine Rule
The cosine rule connects the lengths of the sides of a triangle with the cosine of one of its angles. It is given by:
\[ c^2 = a^2 + b^2 - 2ab \cos C \] where c is the side opposite angle C.
This formula is ideal when you know two sides and the included angle (SAS) or all three sides (SSS) and want to find an unknown side or angle. It generalises Pythagoras’ theorem to any triangle, including those with obtuse angles.
- Cosine rule formula: \( c^2 = a^2 + b^2 - 2ab \cos C \)
- Used for SAS and SSS cases
- Can find unknown side or angle
Recognising and Resolving Ambiguous Cases
When using the sine rule in SSA cases (two sides and a non-included angle), there can be zero, one, or two possible triangles. This is known as the ambiguous case.
To resolve it: calculate the possible value(s) of the unknown angle using the sine rule. If the sine value is greater than 1, no triangle exists. If it is exactly 1, one right-angled triangle exists. If less than 1, two possible angles (one acute and one obtuse) may satisfy the condition, leading to two possible triangles.
Always check the sum of angles to ensure they form a valid triangle (sum must be 180°).
- Ambiguous case arises in SSA configuration
- Check sine value to determine number of solutions
- Verify angle sums to confirm valid triangles
Calculating the Area of a Triangle Using Two Sides and Included Angle
The area of any triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} ab \sin C \] where a and b are two sides enclosing angle C.
This formula is useful when the height is not known or difficult to measure. It applies to all triangles, including obtuse and acute ones.
- Area formula: \( \frac{1}{2} ab \sin C \)
- a and b are sides enclosing angle C
- Works for any triangle shape
Worked Example: Solving a Triangle and Finding its Area
Given triangle ABC where side a = 7 cm, side b = 10 cm, and angle C = 60°, find side c and the area of the triangle.
Step 1: Use the cosine rule to find side c opposite angle C.
\[ c^2 = a^2 + b^2 - 2ab \cos C = 7^2 + 10^2 - 2 \times 7 \times 10 \times \cos 60^\circ \] \[ = 49 + 100 - 140 \times 0.5 = 149 - 70 = 79 \] \[ c = \sqrt{79} \approx 8.89 \text{ cm} \] Step 2: Calculate the area using the formula: \[ \text{Area} = \frac{1}{2} \times 7 \times 10 \times \sin 60^\circ = 35 \times 0.866 = 30.31 \text{ cm}^2 \] Check: Side length and area are reasonable for given data.
- Use cosine rule for side opposite known angle
- Use sine area formula with two sides and included angle
- Always check calculations for consistency
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Using Sine Rule | Relate sides and opposite angles using ratios | Given A=40°, B=60°, a=8 cm, find b: \( b = \frac{a \sin B}{\sin A} = \frac{8 \times \sin 60^\circ}{\sin 40^\circ} \) | Apply when two angles and one side or two sides and a non-included angle are known |
| Using Cosine Rule | Calculate unknown side or angle using sides and included angle | Given a=5 cm, b=7 cm, C=120°, find c: \( c^2 = 5^2 + 7^2 - 2 \times 5 \times 7 \times \cos 120^\circ \) | Use when two sides and included angle or all sides known |
| Ambiguous Case | Check for two possible triangles in SSA case | Given a=7 cm, A=30°, b=10 cm, find possible B values | Always verify number of solutions and angle sums |
| Area Calculation | Use two sides and included angle to find area | Given a=6 cm, b=9 cm, C=45°, area = \( \frac{1}{2} \times 6 \times 9 \times \sin 45^\circ \) | Calculate area without height measurement |
Active recall
Quick check
1. What formula relates the sides and opposite angles in any triangle?
The sine rule: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
2. When is the cosine rule used?
When two sides and the included angle or all three sides are known, to find an unknown side or angle.
3. What is the ambiguous case in triangle solving?
It occurs in SSA cases where two different triangles may satisfy the given conditions.
4. How do you calculate the area of a triangle if you know two sides and the included angle?
Use the formula \( \text{Area} = \frac{1}{2} ab \sin C \), where a and b are the sides enclosing angle C.