Sine Rule (SQA National 5 Maths): Revision Note

Using sine rule to find a length

What is the sine rule?

• The sine rule is used in non right-angled triangles

  • It allows you 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

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

Using sine rule to find an angle

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 calculator, ${\sin }^{-1}\left(...\right)$

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, $\theta$, 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 $\theta$

• The other base angle could either be obtuse or acute

  • After using the sine rule, using sin^-1 on your calculator only gives the acute angle

    • You need to check the diagram or the other information given in the question to see if the angle you need is actually obtuse

    • If it is, use this rule: obtuse angle = 180 - acute angle