The Sine and Cosine Rules

Understanding the Sine and Cosine Rules

Basics of Sine and Cosine Rules

• The sine and cosine rules are fundamental concepts used to solve problems in trigonometry. They can be applied to find unknown sides and angles within non-right angled triangles.

Sine Rule

• The sine rule is a relationship among the lengths of the sides of a triangle and the sines of its angles. It can be expressed in the following way: (a/sinA) = (b/sinB) = (c/sinC).
• It can be used to find an unknown side length if one side and its opposite angle, as well as another angle, are known.
• It can also be used to find an unknown angle if two sides and an angle are known.
  • Remember to check if the unknown angle could be acute or obtuse. Use the fact that sin(180 - θ) = sinθ to check for an obtuse solution.

Cosine Rule

• The cosine rule, also known as the law of cosines, is another relationship among the lengths of the sides of a triangle and the cosine of one of its angles. It can be expressed in the following way: c² = a² + b² - 2ab cosC.
• It is typically used to find unknown side lengths in a triangle when the lengths of two sides and the included angle are known.
• This rule can also be rearranged to find an unknown angle if all three side lengths are known: cosC = (a² + b² - c²) / 2ab.
• As cos(θ) is negative for obtuse angles (90° < θ < 180°), the cosine rule can be applied to determine if an angle is acute or obtuse by calculating the cosine.

Practise using the sine and cosine rules is essential for developing a good understanding. Solving a variety of problems involving non-right angled triangles will help solidify these concepts.