Sine, Cosine Rule & Area of Triangles (AQA GCSE Further Maths)

• Pythagoras' theorem and SOHCAHTOA only work in right-angled triangles

• The Sine Rule and Cosine Rule allow us to answer triangle questions for ANY triangle

• The Area of a Triangle Formula is an alternative to 

Sine & Cosine Rules

What are the sine rule and cosine rule formulae?

• Before applying these formulas it is useful to label the triangle using a system of 'opposites'

• Note how lowercase letters are used for side lengths and capital letters for angles

  • make sure an angle and the side opposite it have the same letter

• Sine Rule

• sin 90° = 1 so if one of the angles is 90°, this becomes ‘SOH’ from SOHCAHTOA

• The sine rule can also be 'flipped over'

• This is more useful when we are using the rule to find angles

• Cosine Rule

• cos 90° = 0 so if A = 90°, this becomes Pythagoras’ Theorem

• The cosine rule is 'cyclic' so there are two other versions of it

• but once you are used to using cosine rule, there's only really one version

  • you just need to 'cycle' a, b and c around accordingly for a specific question

• Note that it's always the angle between the two sides in the final term

• Questions can become algebraic when side lengths are replaced by letters and expressions

  • Use the same rules but put brackets around expressions like (x + 3)

What is the ambiguous case of the sine rule?

• In some circumstances, it is possible to get two possible answers using sine rule

  • this arises when the angle you are trying to find could be either acute or obtuse

• Consider the triangle below

• You know the pair a and A and half of the pair c and C (c = 34)

  • However there are two possible locations for the third vertex of the triangle (labelled C₁ and C₂)

  • So there are two possible angles for C (x and y on the diagram)

• (Most) calculators will only give one answer when using inverse sine ()

  • this will be the acute angle

  • if the obtuse angle is required, subtract the acute angle from 180°

• Questions will state, or imply, whether the acute or obtuse angle is required

  • or could ask you to find both possibilities!

• There is no 'ambiguous' case for the cosine rule

How do I know whether to use sine rule or cosine rule?

• Sine rule is used when you know a 'pair' and half of the pair involving the unknown we are asked to find

  • e.g.  a question may give us side a and angles A and B, then ask us to find side b
    (you know the pair a and A, and half of the pair b and B)

• Cosine rule is used in two circumstances

  • when questions gives us an angle and the two sides that make it

    • this allows us to find the side opposite the given angle

    • e.g.  a question may give us sides b, c and angle A
      (this would allow us to find the side a)

  • when questions give us all three sides of a triangle and we are required to find an angle

    • in this case the rearranged version of cosine rule may be helpful

• then use inverse cos () to find A

• you may prefer to substitute known values into the 'regular' version of cosine rule and then rearrange

Area of a Triangle

What is the area of a triangle formula?

• The Area of a Triangle formula applies to any triangle set up using the system of 'opposites'

• As with cosine rule, there are 'cyclic' alternatives to this

  • but rather than rememeber three formulae, remember one and change a, b and c accordingly

  • e.g.  

• sin 90° =1 so if C = 90°, this becomes 

• In genreal  allows us to find the area without having to know or find the height

  • it is particularly useful in non-right angled triangles

• The questions encountered can be more algebra-based when side lengths are given as letters or expressions

  • use brackets around expressions like (x + 3)

  • check for any exact trigonometric values, like

Applications of Trigonometry

How do I choose which rule or formula to use

• It is important to be able to decide which Rule or Formula to use to answer a question

• This table summarises the possibilities:

What problems will I be asked to solve using trigonometry?

• Problems in trigonometry may involve any, or more than one

  • Pythagoras' theorem

  • SOHCAHTOA

  • Sine rule

  • Cosine rule

  • Area of a triangle

• Beyond 2D triangles you may be asked to solve problems involving

  • more complicated 2D shapes that have parts or sections that can be considered as a triangle

    • e.g. a trapezium

  • 3D shapes that have triangular parts or features

    • e.g. a triangular prism, the cross section of a cone

• Many problems will involve algebraic answers

  • sides may be given in terms of a variable such as x

  • you may be asked to find an expression for a length or area "in terms of x"

• Problems may involve surds, arising from exact values to common trigonometric ratios

  • e.g. 

  • You need to learn these exact values