Use of Sine and Cosine Formulae

Sine and Cosine Formulae

Sine Rule:

The sine rule is applied when we have either two angles and one side of a triangle (AAS or ASA), or two sides and a non-included angle (SSA).
The sine rule states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides of the triangle. This can be written as: a/sinA = b/sinB = c/sinC
This rule can be rearranged to find either a side (use a = b.sinA/sinB) or an angle (use A = sin⁻¹[a.sinB/b]).

Cosine Rule:

The cosine rule is used when we have either three sides (SSS) or two sides and the included angle (SAS) of a triangle.
The cosine rule states that the square of one side of a triangle is equal to the sum of the squares of the other two sides, minus twice the product of these two sides multiplied by the cosine of the included angle. This can be written as: c² = a² + b² − 2ab.cosC
This rule can also be rearranged to find an angle, given the lengths of all three sides: C = cos⁻¹[(a² + b² - c²) / 2ab]

Area of a Triangle:

The formula for the area of a triangle can be derived from the sine rule.
The area A of a triangle with sides of lengths a, b and c and with corresponding angles A, B and C, is given by: A = 0.5ab.sinC

Using Charts or Calculators with Sine or Cosine:

When using charts or calculators, be sure to check if the input should be in degrees or radians.
Calculate angles using sin⁻¹, cos⁻¹, or tan⁻¹ (known as ‘inverse-tangent’). For example, if A = sin⁻¹(0.5), angle A is the angle whose sine is 0.5.
When using the sine and cosine rules, always double-check your work to avoid common errors, like confusing opposite and adjacent sides, or misplacing values in a formula.