• From: Aaron Burgess
  • Date: 17 Feb 1999
  • Subject: Calculating the length of a side with trigonometry

Our teacher set us this problem:

A triangle A B C has the following dimensions:
AB = 18cm,   AC = 12cm,   and BC = 15cm.
What is the size of angle BAC ?

I used tan with BC over AC and got the answer of 51.34°.
But when I check it with sin or cos I get two different answers.

Why is Trigonometry not working?

Maths Help suggests:

Sketch of triangle ABC

If you have a glance at a sketch of the triangle, and study the question, you will notice that the triangle is not necessarily right-angled.

It is important not to assume that the triangle has a right angle, unless you are specifically told it is.

The trigonometrical ratios sin, cos and tan,
and Pythagoras' Rule, only work for
Right-Angled Triangles.

In this case, you must use the Cosine Rule:

This re-arranges to

and substituting a=15, b=12, c=18 leads to the answer 55.8° (to 1 d.p.)

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