Solving Right Triangles

To solve a right triangle means to calculate all unknown lengths and angles.

Now, every triangle (including right triangles) have three angles and three sides, for a total of six parts (as they are called). To solve a triangle, you need to know three of the six parts, and the three known parts must include the length of at least one side.

Since we always know at least one angle of a right triangle (the right angle itself), then the minimum additional information needed to solve a right triangle is either

(i) the length of two of the three sides (called the ‘ss’ case)

or

(ii) the length of one side and the value of one of the acute angles (called the ‘sa’ case)

Some books give formal strategies for each type of right triangle problem. Here, we just advise the following procedure:

(i) list the values of the parts that are known (you don’t need to list the right angle)

(ii) list the symbols for the parts which are unknown and must be calculated.

(iii) then, use Pythagoras’s Theorem and/or the definitions of the principal trigonometric functions and the inverse trigonometric functions to calculate each of the unknown parts, one-by-one. It is a good strategy to set up your calculations so that they use just the original known values given in the statement of the problem. Then if you make an arithmetic error, it doesn’t affect other calculations.

 

Example 1

Solve the right triangle shown in the sketch.

solution

We are given a = 31 and c = 42 and so we must determine b, A, and B.

When two sides of a right triangle are given, the third side can always be determined using Pythagoras’s Theorem: b² = c² – a² = 42² – 31² = 803.

Thus rounded to two decimal places.

Then so

Finally, and so

Thus the required solution is b ≈ 28.34, A ≈ 47.57º, and B ≈ 42.43º.

Notice how we organized our work in this example so that all calculations were based only on information given in the original problem. None of the calculations involved numbers we had previously calculated ourselves. This is a good strategy, because then if we had made an arithmetic error in one step of the solution, it wouldn’t affect the correctness of results of other calculations. This sort of defensive strategy is always possible when solving right triangles.