Trigonometry – Non-Right-Angled Triangles

Using the Sine Rule

Finding Angles with the Cosine Rule

Finding Lengths with the Cosine Rule

Area of a Non-Right-Angled Triangle

Sine Rule

Cosine Rule

Prerequisite Knowledge

• Know the trigonometric ratios Sinϑ = Opp/Hyp, Cosϑ = Adj/Hyp and Tanϑ = Opp/Adj.
• Apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures

Success Criteria

• Know and apply the sine rule [latex]\frac { a }{ SinA } =\frac { b }{ SinB } =\frac { c }{ SinC } [/latex] and cosine rule a² = b² + c² – 2bcCosA to find unknown lengths and angles.
• Know and apply the formula for the area of a triangle [latex]A=\frac { 1 }{ 2 } abSinC[/latex] to calculate the area, sides or angles of any triangle.

Key Concepts

• The Sine rule is used when:
  • Any two angles and a side is known.
  • Any two sides and an angle is known
• The Cosine rule is used when:
  • all three sides are known
  • two sides and the adjoining angle is known
• Students should have the opportunity to derive the three formulae from first principals.
• This topic is often linked with problems involving bearings and map sketches.

Common Misconceptions

• Students often have difficulty choosing the correct formula.
• A common mistake is attempting to use Pythagoras’ Theorem to find a length in a non-right angled triangle.
• Marks are often lost when breaking down the Cosine Rule using the order of operations.