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

What is the ambiguous case of the sine rule?

• If the sine rule is used in a triangle?given two sides and an angle which is not the angle between them?there?may?be more than one possible triangle which could be drawn
• The side?opposite?the given angle could be in two possible positions
• This will create two possible values for each of the missing angles and two possible lengths for the missing side
• The two angles found?opposite?the given side (not the ambiguous side) will?add up to 180°
  • In IB the question will usually tell you whether the angle you are looking for is?acute?or?obtuse
  • The sine rule will always give you the acute option but you can?subtract from 180° to find the obtuse angle
  • Sometimes the obtuse angle will not be valid
    • It could cause the sum of the three interior angles of the triangle to exceed 180°

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
• • C?is the angle?between?sides?a?and?b
• Substitute the values you have into the formula and solve

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 = ? × base × height