SOLVING THE RIGHT TRIANGLE

To "solve a right triangle" means to find all of the missing parts of a triangle with a 90 degree angle in it. In one case, you'll be given a side and an angle. In another case, you'll be given two sides. We consider these two cases.

GIVEN: One known side and one known angle (in addition to the 90 degree angle). Solve the Right Triangle

Example: Given a right triangle with m/B = 23o10', m/C = 90o, and measure of side a = .0345, Solve the right triangle.

STEPS TO FOLLOW:

1. First find the missing angle. Since all of the angles of a triangle add up to 180 degrees and you already have a 90 degree angle (right triangle), you can subtract the known angle measure from 90 to get the measure of the missing angle.

In this example, A = ? a = .0345 B = 23o10' b = ? C = 90o c = ?

(Unknown Angle A) = 90 - (Known Angle B) A = 90 - (23o10')

To use your calculator, you must convert 23o10' to a mixed number. In every 1 degree, there are 60 minutes ( 1o = 60'). 10' means that we have 10 of the 60 minutes in 1 degree. So 10' = 10/60 of a degree. 23o10' means we have 23 + (10/60) degrees. So the equation above becomes, A = 90 - (23o10') = 90 - (23 + (10/60) ) = 66.8333 degrees

To convert 66.8333 into degrees and minutes, multiply 60 by the decimal portion of the answer, (.8333)(60) = 49.998 which is approximately 50 minutes. Thus, 66.8333 = 66o50'. 2. Pick an unknown side. Call it "x" or use the letter of the side. The convention for labeling a triangle is to use the same letter for the side that is opposite a given angle. A lower case letter is used for the side, a capitol letter for the angle. The side opposite angle B, for example, would be labeled "b"; the side opposite angle C would be "c"; and the side opposite A is side "a".

Let's start with side "b" since its the first side missing in our example.

3. Place the letter from Step 2 on top of a fraction. Place the measure of the other known side on the bottom of the same fraction. This creates a trigonometric "ratio" that we will soon use.

So we have the fraction:

4. Pick an angle, but not the 90 degree angle. If the top of the fraction you formed is the letter "b", use angle B. If "a", use angle A. If the top of the fraction is "c", you can use either angle A or B. Next, decide on a trig function to use with this angle.

How to decide which trig function to use: a). What is the relationship (opposite, adjacent, or hypotenuse) of the unknown side on top of the fraction to the angle you have picked? b). What is the relationship of the known side (on the bottom of the fraction) to the angle you have picked? c). Based on the relationships of the sides, choose from:

In our example, "b" is the side opposite angle B (23o 10') "a" ( =.0345) is the side adjacent to angle B. From the list above, opp/adj means we use the tangent function. 5. Set up and solve the trig equation.

In our example,

Repeating Steps 2 -5 of the process for side "c":

2. Pick side "c" as the unknown side.

3. Form fraction:

4. Since side "c", use either Angle A or B. I'll use A ( = 66.8333o) "c" is the hypotenuse and ".0345" is the measure of side a, the side opposite angle A. From the list, hypot/opp is the csc(A).

5. Set-up and solve the equation:

All of the missing pieces have been found. We have solved the Right Triangle:

A = 66o50 a = .0345 B = 23o10' b = .0148 C = 90o c = .0375