The Sine Rule: when to use it and how to do the hard questions
Learn the sine rule by heart and understand each part in depth
The Sine Rule: for any triangle that is NOT a right angled triangle:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Where:
- \( a, b, c \) are the sides of the triangle, or lengths.
- \( A, B, C \) are the angles OPPOSITE those sides.
So, firstly, make sure you label your triangle correctly! Angle A is opposite side a and so on:
When can you use Sine rule?
If you look at the fractions for Sine rule, you will only need 2 of the fractions the a/sin A and b/sin B.
There are 4 unknowns in those 2 fractions.
2 angles and 2 side lengths that are opposite.
You can only solve an equation where one thing is unknown.
This means if you have a question that requires sine rule, the question has to give you the following information:
2 side lengths and an angle that is opposite one of those sides
Or
2 angles and a side length that is opposite one of those angles.
It won’t work if you have 2 side lengths and the angle in between them - that’s cosine rule.
It won’t work if you only have 3 side lengths - again that is cosine rule.
2 side lengths and one opposite angle allows you to calculate the other angle opposite the other known side
If the question gives you 2 angles and 1 side length that is opposite one of the given angles, you can calculate the other side length that is opposite the other given angle
Sine Rule Quiz
1. In a triangle, A = 35°, B = 60°, and side a = 8 cm. Find side b.
Show Hint
Use: \( \frac{a}{\sin A} = \frac{b}{\sin B} \)
2. Find angle C if a = 9 cm, A = 42°, and c = 11 cm.
Show Hint
Rearrange the sine rule to solve for \( C \).
3. A triangle has sides a = 7 cm, b = 12 cm, and angle A = 50°. Find angle B.
Show Hint
Use \( \sin B = \frac{b \cdot \sin A}{a} \).
4. Given A = 48°, a = 10 cm, and b = 14 cm, find angle B.
Show Hint
Find \( \sin B \) using the sine rule and then use inverse sine.
5. In a triangle, A = 72°, B = 40°, and side a = 15 cm. Find side b.
Show Hint
Use: \( \frac{a}{\sin A} = \frac{b}{\sin B} \).
More questions and explanations coming soon!
