Sine rule and cosine rule (Higher Tier)
Sine rule and cosine rule (Higher Tier)
The Sine Rule
Basic Concepts
- The Sine Rule relates the ratio of the lengths of sides of a triangle to the sines of its angles.
- This rule can be used in any triangle, not just right-angled ones.
- The formula for the Sine Rule is a/sinA = b/sinB = c/sinC.
Applications
- The Sine Rule is primarily used to calculate missing side lengths or angles in a triangle.
- To find a missing side length, rearrange the formula to a = b*(sinA/sinB).
- To find a missing angle, rearrange the formula to A = arcsin[(a*sinB)/b].
- The Sine Rule can be used to find an angle when two sides and a non-included angle are known, or to find a side when two angles and a non-included side are known.
The Ambiguous Case of the Sine Rule
- The Ambiguous Case occurs when the Sine Rule is used to find an angle and there are two possible solutions.
- This happens when the length of a side opposite an obtuse angle is less than the length of another side of the triangle.
The Cosine Rule
Basic Concepts
- The Cosine Rule also relates the lengths of sides of a triangle to the cosine of one of its angles.
- It can also be used in any triangle, but is often used when the triangle is not right-angled.
- The formula for the Cosine Rule is c² = a² + b² - 2ab*cosC.
Applications
- The Cosine Rule is used to calculate a missing side length when two sides and the included angle are known.
- It can also be used to calculate an angle when all three sides of the triangle are known.
- To find a missing angle using the Cosine Rule, rearrange the formula to C = arccos[(a² + b² - c²) / (2ab)].
Remember, practicing these rules with different triangles will help these concepts become more familiar. Continue to work through problems and challenge your understanding of these principles.