In previous sections you have learned how to use trig ratios to help solve some geometric problems involving right angled triangles. Now we’ll consider some methods that work also for more general triangles.
After completing this section you should be able to:
1. Read section 7.1 of the text. On your first reading of section 7.1, skip from the end of Example 1 directly to Example 4. After reading section 7.1 try Exercise #17, and check your answer (and if necessary refer to the student solutions manual to see how it’s done).
2. Read section 7.2 up to the end of Example 6. (You can also read the rest if you like but you will not be expected to remember it). Try Exercise #13 if you want to see an applied example (and check the solutions manual if you have difficulty).
3. Read the following Summary Notes and Discussion, and
4. Follow the instructions regarding Further Practice.
Note that the picture used to prove the Law of Sines in the text (Figure 1 on page 520) has the same pattern as the one for answer #1 in our previous section. In fact, any problem that can be solved using the Law of Sines or Law of Cosines could also be done by just adding extra construction lines to make right triangles, but it is certainly often more convenient to have these rules as a shortcut.
The Law of Sines is most useful for solving triangles given one side and two angles (it doesn’t matter which two angles are given, since once we know two we can find the third by subtraction).
The Law of Cosines works whenever we are given two sides and the contained angle (SAS), but if we are given two sides and just one of the other two angles it is more difficult. In fact for this (SSA) case there is not always a triangle that fits the given data, and for some data there are more than one.
It is worth noting that the Law of Cosines can also be applied in the SSA case. If b is the unknown side and angle at A is the known angle, then is a quadratic equation for b, and the “ambiguous case” is just when this equation has two distinct positive roots.
How to deal with SSA cases using the Law of Sines is discussed in the text material in section 7.1 following Example 1 and up to the end of Example 3. You should read this material after you have done enough pratice to get comfortable with the more straightforward cases.
Check your understanding and practice for speed by working through some of the Exercises on pp537-540 and pp546-550.
Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. (Note that, as usual, the answers are in the back of the text and complete worked solutions are in the student study guide - but try to avoid looking at answers or solutions until you have made your own best effort)
As a minimum, start
with ##21 and 23 of sec7.1 and ##19 and 25 from sec 7.2.