The Sine Rule

Starter a) Calculate the Area of an Equilateral triangle with a Perimeter of 30cm (Do this using Trigonometry, not Pythagoras!) b) What about if the side length was x?

The Sine Rule The Trigonometry we have seen already, involving ‘SOHCAHTOA’ works in Right-angled triangles There are also some patterns that will work in any triangle They are called the ‘Sine Rule’ and the ‘Cosine Rule’ Today we will be looking at the Sine Rule, which you are given in the exam booklet

The Sine Rule Learning Objectives All will be able to use the Sine rule to find missing sides in triangles (Grade A) Most will also be able to use it to find missing angles as well (Grade A) Some will be able to use it in answering longer questions! (Grade A/A*)

The Sine Rule B Consider the triangle labelled to the right, remembering GCSE trigonometry: Right hand triangle: Left hand triangle: The opposite sides are the same so: hyp hyp c a opp A C adj b adj S O H Divide by c and a 2A

The Sine Rule In any triangle: OR a SinA b SinB c SinC = = SinA a SinBb SinC c = =

The Sine Rule Work out the value of the missing side, labelled ‘a’ a SinA b SinB c SinC Work out the value of the missing side, labelled ‘a’ = = a Sin74 12 Sin49 = 49° a 12 Sin49 74° a = x Sin74 12cm a = 15.2842… a = 15.28 (2dp)

The Sine Rule Work out the value of the missing side, labelled ‘y’ a SinA b SinB c SinC Work out the value of the missing side, labelled ‘y’ = = a Sin101 8.1 Sin38 = a 8.1m 8.1 Sin38 a = x Sin101 101° 38° a = 12.9148… a = 12.9 (1dp)

The Sine Rule Work out the value of the missing angle, θ. Sinθ 16 SinA a SinB b SinC c Work out the value of the missing angle, θ. = = Sinθ 16 Sin 76 18 = 18mm 16mm Sin 76 18 Sinθ = x 16 θ 76° Sinθ = 0.86248… Sin-1 θ = 59.6° (1dp)

Plenary a SinA b SinB c SinC = = AB Sin24 17 Sin95 = 17 Sin95 AB = x 6.9409… 6.94

Plenary Learning Objectives All will be able to use the Sine rule to find missing sides in triangles (Grade A) Most will also be able to use it to find missing angles as well (Grade A) Some will be able to use it in answering longer questions! (Grade A/A*)

Summary We have learnt another rule of Trigonometry, the Sine Rule We have used it to calculate missing angles and sides in a triangle We have also seen that it works in any triangle, not just right-angled ones…