Title: Find all solutions in :
Author: Mary R. Parker

Trigonometry Problems:

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: 

  

Trigonometry problem comments and answers:  These comments should give enough clues so that you could look up explanations and similar problems in a trig book.  

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 1 here): 
4.  Graphing trig functions (rewrite it as ):  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 here.  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.