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# Solving Right Triangles

Example 1

Solve the right triangle sketched below.

solution

We are given A = 27Âº and b = 412 m

To solve this triangle, we need to compute values for B, a, and c.

When one of the acute angles is given, the easiest way to compute the other is by subtraction from 90Âº:

A + B = 90Âº (for a right triangle)

So, 27Âº + B = 90Âº giving B = 90Âº - 27Âº = 63Âº.

Then, for side a, we have or which gives a = (412 m)(tan 27Âº) ≈ 209.92 m.

Finally, for side c, or giving

Thus, the required solution is: B = 63Âº, a ≈ 209.92 m, c ≈ 462.40 m

Note the strategy employed in the previous example. To calculate â€˜aâ€™, we looked for a trigonometric ratio that involved A and b (our given quantities) and â€˜aâ€™ (the unknown we wished to determine). Similarly, to determine c, we looked for a trigonometric ratio involving A, b, and c. In both of these cases this meant that the definition of the selected trigonometric ratio would amount to an equation with one unknown, which was then easily solved.

Example 2

Solve the right triangle shown in the figure below.

solution

The solution here is that there is no solution. Knowing all three angles of a right triangle is not enough information to be able to calculate the lengths of any of the three sides. In fact, there is an infinite number of right triangles that have these three angles â€“ each of a different size.

So, we cannot obtain a unique solution for this triangle.

Example 3

Solve the right triangle with c = 26.8 cm and B = 37.56Âº.

solution

 It is probably best to start by making a sketch, as is done to the right. In addition to the right angle, of course, the known parts of the triangle, B and c, are given in the statement of the problem. So, we need to determine the values of A, a, and b. First, A + B = 90Âº so A + 37.56Âº = 90Âº giving A = 90Âº - 37.56Âº =52.44ÂºThen, gives or b = (26.8 cm)(sin 37.56Âº) ≈ 16.34 cm Finally, using or gives us a = (26.8 cm)(cos 37.56Âº) ≈ 21.24 cm Thus, the required solution is A = 52.44Âº, a ≈ 21.24 cm, b ≈ 16.34 cm.