Sine & Cosine Formulae (Edexcel IGCSE Further Maths)

Sine Formula

Spec point — 10D

What is the sine formula?

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

  • It is also known as the sine rule

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

  • $\frac{a}{\mathrm{sin}A}=\frac{b}{\mathrm{sin}B}=\frac{c}{\mathrm{sin}C}$

• a is the side opposite angle A

• b is the side opposite angle B

• c is the side opposite angle C

• This formula is not on the exam formula sheet, so you need to remember it

• Remember that sin 90° = 1

  • So if one of the angles is 90° this becomes SOH from SOHCAHTOA

How can I use the sine formula to find missing side lengths or angles?

• The sine formula 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 as given above to find the length of a side

• To find a missing angle it's easier to 'flip' the formula

  • $\frac{\mathrm{sin}A}{a}=\frac{\mathrm{sin}B}{b}=\frac{\mathrm{sin}C}{c}$

• Substitute the values you have into the formula and solve

  • This will always give you an acute angle
  • If you know the angle is obtuse then subtract this value from 180

Exam Hint

• Remember to check that your calculator is in the correct angle mode (degrees or radians)!

Worked Example The following diagram shows triangle $ABC$. $AB=8.1\mathrm{cm}$, $BC=12.3\mathrm{cm}$, and angle $BCA=27^{\circ}$. It is also given that angle $BAC$ is acute.

(a) Find the value of $x$,

(b) Find the value of $y$.

Cosine Formula

Spec point — 10D

What is the cosine formula?

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

  • It is also known as the cosine rule

• It states that for any triangle

  • $a^{2}=b^{2}+c^{2}-2bc\mathrm{cos}A$

• a is the side opposite angle A

• b and c are the other two sides

• angle A is the angle in between sides b and c

• This formula is on the exam formula sheet

  • So you don't need to remember it
  • But you do need to be able to use it

• Remember cos 90° = 0

  • So if A = 90° this becomes Pythagoras’ Theorem

How can I use the cosine formula 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

• Use the formula in the formula sheet form to find an unknown side

• Rearrange the formula to find an unknown angle

  • $\mathrm{cos}A=\frac{b^{2}+c^{2}-a^{2}}{2bc}$

    • Remember, A must be the angle in between sides b and c

• Substitute the values you have into the formula and solve

Exam Hint

• Remember to check that your calculator is in the correct angle mode (degrees or radians)!

Worked Example The following diagram shows triangle $ABC$. $AB=4.2\mathrm{km}$, $BC=3.8\mathrm{km}$, and $AC=7.1\mathrm{km}$.

Calculate the size of angle $ABC$.

Area of a Triangle

Spec point — 10D

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

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

  • $\mathrm{Area}=\frac{1}{2}ab\mathrm{sin}C$

    • C is the angle in between sides a and b

  • This formula is not on the exam formula sheet, so you need to remember it

• Remember sin 90° = 1

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

• If you know the area you can use the formula to find the length of a side or an angle

  • Using inverse sin will always give you an acute angle

    • Subtract the value from 180 if you know the angle is obtuse

Exam Hint

• Be careful to label your triangle correctly

  • so that your angle C is always the angle between the two sides

• Remember to check that your calculator is in the correct angle mode (degrees or radians)!

Worked Example The following diagram shows triangle $ABC$. $AB=32\mathrm{cm}$, $AC=1.1m$, and angle $BAC=74^{\circ}$.

Calculate the area of triangle, giving your answer correct to 3 significant figures.