and then find its value. Review of Trigonometry
Download a Microsoft Word file or a Rich Text File (works on most word processors) with the problems and answers
Students in scientific calculus are expected to have a solid foundation in trigonometry. One of the problems with this is that trigonometry questions occupy a fairly small amount of time in every course after the trig course. Students forget what they have learned when they don't use it often and then there is never really enough time in any later course to use it for long enough at any one time to thoroughly refresh the topics. I suggest that any student in a precalculus or calculus course look over this set of problems and remember how to do them at least once each semester. Obviously, the point is not to learn to do only these problems, but also to remember how to do similar problems of each type. In calculus, there will be quite a lot of solving trig equations, so be sure to focus on that.
A list of the facts that you should have memorized in your trigonometry class is available on the Web. You may not remember all of them now, but, before you start using trigonometry in a later course, you should review these and memorize them again.
1. Give at least two other ways to write the expression and then find its value.
2. Find all solutions in 
: 
3. Solve 
in the interval 
4. For the following trig function, graph it, including find the amplitude, period, phase shift, and x intercepts: 
.
5. Explain the difference between 
and 
.
6. Find the length of a guy wire that makes an angle of 
with the ground if the wire is attached to the top of a tower 63.0 meters high.
7. Solve the triangle ABC (not a right triangle) if A
, b=12.9 meters, and c=15.4 meters.
8. Use an appropriate half-angle identity to find the exact value of 
.
9. Prove the following identity: 
1. Trig functions: 
. The main point here is to notice what the squared means, and the secondary point is to be able to interchange degree and radian measure for angles. 
. (Actually, most students will be able to do this problem, but may make mistakes when they have variables rather than numbers, perhaps thinking that 
or 
, both of which are incorrect.)
2. Radian measure and circular functions, trig equations: 
3. Trig equations: Get 0 on one side, factor, solve the resulting two equations as in number 2 in this problem set. Solutions: 
4. Graphing trig functions: Rewrite it as 
. The amplitude is 2, the minus in front means it is reflected across the y axis, the period is 
, and the x intercepts are 
to the left of those for the regular cos function, which are 
etc.
5. Inverse trig functions: The first here is the inverse under composition, which is 
. It has domain all real numbers. The second here is 
. It has domain all real numbers except integer multiples of 
. You can further emphasize the difference between the two by evaluating them at several values of x and showing that you get very different values and by graphing them. Notice that the exponent of -1 is not treated like the exponent of 2 in problem number 1 in this problem set. The reason is that, for functions, we need a notation for the inverse function under composition, and the first notation here is used for that. For numbers, we don't have the composition operation, so that we can use these two notations interchangeably. In fact, this potential for confusion is part of the reason why we have the different names for the reciprocals for the trig functions, so mathematicians and scientists would not use the notation 
, but would use 
instead. In older books, you would not find the notation 
used nearly as much as 
. However, in newer books, with our increased emphasis on function notation, the notation 
is quite common. All the same considerations apply to the notation for the inverse functions of the other trig functions.
6. Solving right triangles: Sketch a diagram with the guy wire as the hypotenuse of a right triangle, set up an equation with the sin function, and then solve for the unknown value of the hypotenuse. length 
meters
7. Law of Cosines and Law of Sines: Start by finding side a using the law of cosines and we find it to be 10.5 meters. There are several approaches from here. One is to find the law of sines to find one of the other angles. Notice that B must be smaller than angle C, since side b is shorter than side c. Since a triangle can't have two obtuse angles in it, then B, since it is smaller, must be acute. That enables us to avoid any ambiguity in using the law of sines to find B
. Using the fact that the three angles must sum to 
, we find that angle A must be 
.
8. Trig Identities: Express 
as half of 
, use the exact value for 
, and plug this into the half-angle identity for cosine. The answer is 
.
9. Trig identities: Work with the left side, write everything in terms of 
, simplify, rewrite this using a basic identity, turn it into an expression with 
in the denominator, and then rewrite that using a basic identity to get the right-hand side.
This page was prepared by Mary Parker. While it has been proofread, it may still have errors. Please send me a message if you find any errors, so that I can correct them! It was last updated January 10, 1999.