Practice Law of sines
The sine rule relates each side of a triangle to the sine of its opposite angle.
It helps you calculate missing sides or angles beyond right-angled triangles.
Exercises with solutions
The sine rule states a ÷ sin α = b ÷ sin β = c ÷ sin γ. It holds in every triangle and links each side to the sine of the opposite angle. You always need a complete pair of side and opposite angle; the sum of angles gives you the third angle.
Exercise 1
In a triangle a = 6 cm, α = 40° and β = 60°. Calculate b.
Show solution and working
b ≈ 8.08 cm
b = a · sin β ÷ sin α = 6 cm · sin 60° ÷ sin 40° ≈ 6 cm · 0.866 ÷ 0.643 ≈ 8.08 cm.
Exercise 2
In a triangle a = 5 cm, α = 30° and γ = 80°. How large is β?
Show solution and working
70°
The angle sum is 180°: β = 180° − 30° − 80° = 70°.
Exercise 3
When is the sine rule the simplest route?
Show solution and working
when a complete pair of side and opposite angle is known
As soon as one side and its opposite angle are known, any other side can be found with the sine rule.
Exercise 4
In a triangle b = 8 cm, β = 50° and γ = 70°. Calculate c.
Show solution and working
c ≈ 9.81 cm
c = b · sin γ ÷ sin β = 8 cm · sin 70° ÷ sin 50° ≈ 8 cm · 0.940 ÷ 0.766 ≈ 9.81 cm.
Common mistake
Inserting the sine of the wrong angle. In the sine rule each side always pairs with its opposite angle, never with an adjacent one. And overlooking that the angle sum provides the third angle.
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