Non Right-Angled Trigonometry (DP IB Maths: AI HL)

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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

begin mathsize 22px style fraction numerator a over denominator sin space A blank end fraction equals blank fraction numerator b over denominator sin space B end fraction equals blank fraction numerator c over denominator sin space C blank end fraction end style

    • 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

Non-Right-Angled Triangles Diagram 1a

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

fraction numerator sin space A blank over denominator a end fraction equals blank fraction numerator sin space B blank over denominator b end fraction equals blank fraction numerator sin space C blank over denominator c end fraction

    • 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

Examiner Tip

  • If you're using a calculator make sure that it is in the correct mode (degrees/radians)
  • Remember to give your answers as exact values if you are asked too

Worked example

The following diagram shows triangle ABC.  AB space equals space 8.1 space cm, AC space equals space 12.3 space cm, straight B straight C with hat on top straight A equals 27 degree.

3-3-2-sine-rule-we-question

Use the sine rule to calculate the value of:

i)
x,

 3-3-2-ai-sl-sine-rule-we-solution-i

ii)
y.

3-3-2-ai-sl-sine-rule-we-solution-ii

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Ambiguous Sine Rule

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°

aa-sl-3-3-2-ambiguous-sine-rule-diagram-1

Examiner Tip

  • Make sure that you are clear which of the two answers is the one that is required and make sure that you communicate this clearly to the examiner by writing it on the answer line! 

Worked example

Given triangle ABC, AB space equals space 8 space cm, BC space equals space 5 space cm, straight B straight A with hat on top straight C equals 35 degree.  Find the two possible options for straight A straight C with hat on top straight B, giving both answers to 1 decimal place.

aa-sl-3-3-2-ambiguous-sine-rule-we-solution

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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

size 22px c to the power of size 22px 2 size 22px space size 22px equals size 22px space size 22px a to the power of size 22px 2 size 22px space size 22px plus size 22px space size 22px b to the power of size 22px 2 size 22px space size 22px minus size 22px space size 22px 2 size 22px a size 22px b size 22px cos size 22px C   ;     begin mathsize 22px style cos space C blank equals blank fraction numerator a to the power of 2 blank end exponent plus blank b squared minus blank c squared over denominator 2 a b end fraction end style

    • 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 C 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 c is opposite angle C, and so on
  • Use the formula c squared space equals space a squared space plus space b squared space minus space 2 a b cos C to find an unknown side
  • Use the formula cos space C blank equals blank fraction numerator a to the power of 2 blank end exponent plus blank b squared space minus blank c squared over denominator 2 a b end fraction  to find an unknown angle
    • C is the angle between sides a and b
  • Substitute the values you have into the formula and solve

Examiner Tip

  • If you're using a calculator make sure that it is in the correct mode (degrees/radians)
  • Remember to give your answers as exact values if you are asked too

Worked example

The following diagram shows triangle ABC. AB space equals space 4.2 space kmBC space equals space 3.8 space km, AC space equals space 7.1 space km.

3-3-2-cosine-rule-we-question

Calculate the value of straight A straight B with hat on top straight C.

3-3-2-ai-sl-cosine-rule-we-solution-relabelled

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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

A space equals space 1 half a b sin C

    • 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

 

Non-Right-Angled Triangles Diagram 2

Examiner Tip

  • If you're using a calculator make sure that it is in the correct mode (degrees/radians)
  • Remember to give your answers as exact values if you are asked too

Worked example

The following diagram shows triangle ABC. AB space equals space 32 space cmAC space equals space 1.1 space straight m, B A with hat on top straight C space equals space 74 degree3-3-2-area-rule-we-question

Calculate the area of triangle .

3-3-2-ai-sl-area-rule-we-solution

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.