Overview
This lecture explains the Sine Rule for solving triangles, demonstrates how to find missing sides or angles, and discusses the ambiguous case.
The Sine Rule
- The Sine Rule relates the sides and angles of any triangle: ( a/\sin A = b/\sin B = c/\sin C ).
- "a", "b", and "c" are side lengths; "A", "B", and "C" are the angles opposite those sides.
- Usually, only two pairs (a/A and b/B) are needed to solve a problem.
Finding a Missing Side
- Label the side you need to find as "a" and the angle opposite as "A".
- Write the Sine Rule with known values.
- Rearrange to solve for the missing side: ( a = \frac{b \cdot \sin A}{\sin B} ).
- Substitute numbers and use a calculator to solve.
- Round your answer appropriately and include units.
Finding a Missing Angle
- Use the Sine Rule in its reciprocal form: ( \frac{\sin A}{a} = \frac{\sin B}{b} ).
- Substitute known side lengths and angle measures.
- Rearrange to isolate (\sin A), then use the inverse sine function to find angle A.
- Always round the angle as instructed, typically to one decimal place.
The Ambiguous Case (SSA)
- Sometimes, the Sine Rule gives two possible angle solutions: ( \text{Angle} = x ) or ( 180^\circ - x ).
- The correct angle (acute or obtuse) depends on context or extra information given in the problem.
- This is known as the "ambiguous case" and occurs when two triangle configurations are possible.
Key Terms & Definitions
- Sine Rule — Relates the ratio of side length to the sine of its opposite angle in any triangle.
- Ambiguous Case — When given two sides and a non-included angle (SSA), two different triangles may satisfy the given conditions.
- Inverse Sine ((\sin^{-1})) — Used to calculate an angle when its sine value is known.
Action Items / Next Steps
- Practice solving triangle problems using the Sine Rule.
- Review ambiguous case examples and ensure you can identify when it occurs.
- Complete the linked exam questions as assigned.