The Sine Rule (Edexcel IGCSE Maths A (Modular))

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Sine Rule

What is the sine rule?

The sine rule is used in non right-angled triangles
- It allows us to find missing side lengths or angles
It states that for any triangle with angles A, B and C

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

Where
- $a$ is the side opposite angle A
- $b$ is the side opposite angle B
- $c$ is the side opposite angle C

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

How do I use the sine rule to find missing lengths?

Use the sine rule
- when you have opposite pairs of sides and angles in the question
  - a and A, or b and B, or c and C
Start by labelling your triangle with the angles and sides
- Angles have upper case letters
- Sides opposite the angles have the equivalent lower case letter
To find a missing length, substitute numbers into the formula
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
- You only need to have two parts equal to each other (not all three)
  - Then solve to find the side you need

How do I use the sine rule to find missing angles?

To find a missing angle, it is easier to rearrange the formula first by flipping each part
$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$
- The angles are now in the numerators of the fractions
- Substitute the values you have into the formula and solve
  - You will need to use inverse sine in your calculation, $\sin^{- 1} (. . .)$

What is the ambiguous case of the sine rule?

Given information about a triangle, there may be two different ways to draw it
In the diagram below, the lengths of two sides are given, a and b
- A base angle is also given, $θ$ , but no angle near b is given
- It turns out that there are two possible ways to arrange b to complete the triangle!
  - Both triangles have the correct values of a, b and $θ$
To other base angle could either be obtuse or acute
- The sine rule only gives the acute answer on your calculator
  - You need to check the diagram to see if the angle you need is actually obtuse
  - If it is, use this rule: obtuse angle = 180 - acute angle

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

Exam Tip

The sine rule is given in the formula booklet.

Worked Example

The following diagram shows triangle ABC.

$AB = 8.1 cm$ , $BC = 12.3 cm$ and angle $BCA = 27 °$ .

Triangle ABC with AB = 8.1 cm, BC = 12.3 cm, AC = y cm, angle BAC = xº and angle BCA = 27º.

(a) Calculate the value of $x$ .

Label the sides of the diagram

Triangle ABC with sides opposite angles labelled with corresponding lowercase letters.

x is an angle so use the sine rule with the angles on top

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$
In practice, you only need to equate two of these three parts

$\begin{array}{rcl} \frac{\sin x}{12.3} & = & \frac{\sin 27}{8.1} \\ \sin x & = & \frac{12.3 \sin 27}{8.1} \\ x & = & \sin^{- 1} (\frac{12.3 \sin 27}{8.1}) \\ x & = & 43.58207 . . . \end{array}$

$43.6 °$ (to 1 d.p)

(b) Calculate the value of $y$ .

To find y you need to know the angle opposite (angle ABC)
You know 27 and x from above, so subtract these from 180

$\begin{array}{rcl} Angle A B C & = & 180 - 27 - 43.58207 . . . \\ = & 109.41792 . . . \end{array}$

y is a length so use the sin rule with the sides on the top
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

$\begin{array}{rcl} \frac{y}{\sin (109.41792 . . .)} & = & \frac{8.1}{\sin 27} \\ y & = & \frac{8.1 \sin (109.41792 . . .)}{\sin 27} \\ y & = & 16.82691 . . . \end{array}$

$y = 16.8$ cm (to 3 s.f.)

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