Test IV Review Sheet for
Colorado State University, Pueblo
Math 124 — Pre-calculus Math — Spring 2015

Topics

1. basic idea of a trigonometric identity: an equation which changes the look of an expression involving trigonometric functions and other algebraic operations but doesn't change its value
2. the book calls these reciprocal identities but I usually think of them as the definitions of the other trig functions in terms of $\sin$ and $\cos$:
    
   
   • $\tan x = {}$
      
     
   • $\sec x = {}$
      
     
   • $\csc x = {}$
      
     
   • $\cot x = {}$
      
     
   KNOW THESE
3. the three Pythagorean identities
    
   
   • 
      
     
   • 
      
     
   • 
      
     
   KNOW THESE
4. even/odd identities (or, the even and odd properties of the three basic trig functions):
    
   
   • 
      
     
   • 
      
     
   • 
      
     
   KNOW THESE
5. cofunction identities (or, shifting the trig functions left or right, and sometimes flipping across the $y$-axis, to get other trig functions), e.g.:
    
   
   • $\sin x = \cos(x + \ \ \ )$
      
     
   • $\cos x = \sin(x - \ \ \ )$
      
     
   • $\tan x = \cot(\ \ \ - x)$
      
     
   KNOW THESE
6. addition and subtraction formulæ, including
    
   
   • $\sin(x+y) = $
      
     
   • $\cos(x+y) = $
      
     
   KNOW THESE
7. sums of sines and cosines, being:
    
   
   • $A\sin x + B\cos x = k\sin(x+\phi)$, where $k=\sqrt{A^2+B^2}$ and $\phi$ satisfies $\cos\phi = A/k$ and $\sin\phi = B/k$.
      
     
   DON'T MEMORIZE THIS
8. double angle formulæ, being
    
   
   • $\sin 2x = $
      
     
   • $\cos 2x = $
      
     
   DON'T MEMORIZE THESE (you can get them from the addition formulæ very easily)
9. product-to-sum formulæ, including
    
   
   • $\sin u \cos v = $
      
     
   • $\sin u \sin v = $
      
     
   DON'T MEMORIZE THESE
10. sum-to-product formulæ, including
     
    
    • $\sin x + \sin y = $
       
      
    • $\cos x + \cos y = $
       
      
    DON'T MEMORIZE THESE
11. using triangles to work out the algebraic expressions for trig functions composed with inverse trig functions.
12. solving basic trig equations by finding the specific solutions in the first period of the function (usually $[0,2\pi)$ for $\sin$ and $\cos$ and $(-\pi/2,\pi/2)$ for $\tan$) and then making the whole solution set by adding $2\pi n$ for $\sin$ and $\cos$ and $\pi n$ for $\tan$, where $n$ is allowed to be any integer.
13. solving more complicated trig equations by using all of our trig identity techniques along with the usual algebraic stuff like the Zero Product Property, common denominators, factoring, the quadratic formula, finding solutions of polynomial equations (where maybe the role of the variable is played by a $\sin x$ or such).

Format of the test

To get credit for a solution on the test, you must always explain your work;correct answers without explanation usually are worth zero points. In particular, if you are asked to prove or verify an identity, you should take one side of the proposed identity and transform it, a single step at a time, with clear explanation of each step, into the other side. Also, whenever possible, give exact answers — e.g, if the answer to a problem is $\sqrt{2}$, please write that, do not write $1.41421356237309504880168$, no matter how tempting.

Don't forget your calculator, you will need it!