An Example of How to Use These Rules

Given a triangle of side lengths b = 3 and c = 6 and angle A = 20°, find the side length a and the angles B and C.

First of all we can find a using the cosine rule.

a² =3² + 6² - (2 x 3 x 6)cos(20)

⇒ a² =9 + 36 - (36 x cos(20))

⇒ a² =11.17

and a=√11.17 = 3.34 (2 decimal place)

We can then calculate angles B and C. At first glance you might think that you can use the sine rule to find them both, but beware! You have to use the sine rule with care, because the inverse of sin, or the arcsin function, is only valid for angles less than 90°. In this example, angle C is obtuse and therefore greater than 90°, if you applied arcsin you would get an incorrect value.

To avoid this problem, remember that the largest angle in a triangle will always be opposite the longest side, also there can only ever be one obtuse angle in a triangle. So, to be on the safe side don't use the Sine Rule to find the largest angle in a triangle, unless you definitely know it is not obtuse.

In this case, c is longest side so we will not use the sine rule to find C, but we can use it to find B.

So, 3 ÷ sinB = 3.34 ÷ sin(20)

⇒ sinB = 3 x sin(20) ÷ 3.34

⇒ sinB = 0.31

⇒B = 17.88° (2 decimal places)

Finally we can find the final angle C, by using the sum of the internal angles rule.

C = 180 - B - A

⇒ C = 142.12° (2 decimal places)

You can see a screenshot of Math.Trig for this triangle opposite, as well as the results page. Don't forget to check the answers using our free iPhone app, Math.Trig!