Sine Rule, Cosine Rule & Area of a Triangle (DP IB Applications & Interpretation (AI)): Revision Note

How should I label a non-right angled triangle?

  • Label the angles with uppercase letters (e.g. A, B, C)

  • Label each side with the lowercase version of the letter of the opposite angle (e.g. a, b, c)

Non Right-Angled Triangle labelled with angles A, B and C and opposite corresponding sides a, b and c.

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Sine Rule

What is the sine rule?

  • The sine rule connects the angles and side lengths of any triangle

  • The formula is fraction numerator a over denominator sin A end fraction equals fraction numerator b over denominator sin B end fraction equals fraction numerator c over denominator sin C end fraction

Examiner Tips and Tricks

This is given in the formula booklet under the geometry and trigonometry section.

  • The formula can be rearranged to fraction numerator sin A over denominator a end fraction equals fraction numerator sin B over denominator b end fraction equals fraction numerator sin C over denominator c end fraction

How can I use sine rule?

  • You need to know an angle and the length of the side opposite the angle

    • Label these as A and a

If you need to find a missing length (b)

  • Label the angle opposite the length as B

    • Use fraction numerator a over denominator sin A end fraction equals fraction numerator b over denominator sin B end fraction

    • Rearrange to get b equals fraction numerator a sin B over denominator sin A end fraction

If you need to find a missing angle (B)

  • Label the side opposite the angle as b

    • Use fraction numerator sin A over denominator a end fraction equals fraction numerator sin B over denominator b end fraction

    • Rearrange to get sin B equals fraction numerator b sin A over denominator a end fraction

    • Use the inverse sine function B equals sin to the power of negative 1 end exponent open parentheses fraction numerator b sin A over denominator a end fraction close parentheses

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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Cosine Rule

What is the cosine rule?

  • The cosine rule connects the angles and side lengths of any triangle

  • The formula is c squared equals a squared plus b squared minus 2 a b cos C

  • The formula can be rearranged to cos C equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction

Examiner Tips and Tricks

Both are given in the formula booklet under the geometry and trigonometry section.

How can I use cosine rule?

If you need to find a missing length (c)

  • Label the angle opposite the length as C

    • Use c squared equals a squared plus b squared minus 2 a b cos C

    • Take the positive square root to get c equals square root of a squared plus b squared minus 2 a b cos C end root

If you need to find a missing angle (C)

  • Label the side opposite the angle as c

    • Use cos C equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction

    • Use the inverse cosine function C equals cos to the power of negative 1 end exponent open parentheses fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction close parentheses

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 formula for the area of any triangle is A equals 1 half a b sin C

    • Make sure that the angle is the one formed by the two sides

  • You can rearrange the formula to find a missing side or angle

Examiner Tips and Tricks

This is given in the formula booklet under the geometry and trigonometry section.

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 degree

3-3-2-area-rule-we-question

Calculate the area of triangle .

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

How do I decide which trig rule to use?

  • Different rules are required depending on the question

    • You need to be able to decide which is appropriate to use

    • Think about what information you have and what you want to find

  • This table summarises the possibilities:

If you know

And you want to know

Use

Two sides and an angle opposite one of the sides

The angle opposite the other side

Sine rule

Two angles and a side opposite one of the angles

The side opposite the other angle

Sine rule

Two sides and the angle between them

The third side

Cosine rule

All three sides

Any angle

Cosine rule

Two sides and the angle between them

The area of the triangle

Area of a triangle rule

The area and the two sides

The angle between the two sides

Area of a triangle rule

The area, a side and an angle

The other side that forms the angle

Area of a triangle rule

Flow chart to determine which rule or formula to use.

Can I use multiple trig rules in the same question?

  • Harder questions will require you to use more than one trig rule

    • For example, you may need the sine rule followed by the cosine rule

  • The area formula only works for an angle between two sides

    • If you are not given this setup, you may need to use the sine or cosine rule first

Examiner Tips and Tricks

If it looks like none of the rules work, then remember that all angles in a triangle sum to 180°. This often helps to find a missing angle, which then allows you to use one of the rules.

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