Non Right-Angled Trigonometry (DP IB Applications & Interpretation (AI)): Revision Note

Sine Rule

What is the sine rule?

• The sine rule allows us to find missing side lengths or angles in non-right-angled triangles

• It states that for any triangle with angles A, B and C
  

  
  

  • Where

    • a is the side opposite angle A

    • b is the side opposite angle B

    • c is the side opposite angle C

• This formula is in the formula booklet, you do not need to remember it

• Sin 90° = 1 so if one of the angles is 90° this becomes SOH from SOHCAHTOA

How can we use the sine rule to find missing side lengths or angles?

• The sine rule can be used when you have any opposite pairs of sides and angles

• Always start by labelling your triangle with the angles and sides

  • Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters

• Use the formula in the formula booklet to find the length of a side

• To find a missing angle you can rearrange the formula and use the form
  

  
  

  • This is not in the formula booklet but can easily be found by rearranging the one given

• Substitute the values you have into the formula and solve

Cosine Rule

What is the cosine rule?

• The cosine rule allows us to find missing side lengths or angles in non-right-angled triangles

• It states that for any triangle

  
     ;     

  • Where

    • c is the side opposite angle C

    • a and b are the other two sides

• Both of these formulae are in the formula booklet, you do not need to remember them

  • The first version is used to find a missing side

  • The second version is a rearrangement of this and can be used to find a missing angle

• Cos 90° = 0 so if C = 90° this becomes Pythagoras’ Theorem

How can we use the cosine rule to find missing side lengths or angles?

• The cosine rule can be used when you have two sides and the angle between them or all three sides

• Always start by labelling your triangle with the angles and sides

  • Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters

• As the formula uses for the known angle, or the angle being found, you can choose to relabel the diagram to match this

  • Remember to also relabel the sides, so that side  is opposite angle , and so on

• Use the formula  to find an unknown side

• Use the formula  to find an unknown angle

  • C is the angle between sides a and b

• Substitute the values you have into the formula and solve

Area of a Triangle

How do I find the area of a non-right triangle?

• The area of any triangle can be found using the formula

  

  • Where C is the angle between sides a and b

  • This formula is in the formula booklet, you do not need to remember it

• Be careful to label your triangle correctly so that C is always the angle between the two sides

• Sin 90° = 1 so if C = 90° this becomes Area = ½ × base × height