The next section in chapter 2 is about the graphs of the basic trigonometry functions. I will begin with the book beside me as I make a few online notes.

• "It's really useful to remember what the graphs of the sin, cos, and tan functions look like." [p. 35]

I am familiar with the overall shapes, but am weak at knowing exactly where the graphs cross the x-axis.

• "You should be able to produce this graph without thinking, including the positions of 0, p/2, p, 3p/2, and 2p" [p. 35]
  

Got it.

• Use the graph of sin(x) to determine values of x that are multiples of p or p/2" [p. 35]
  
  
• "Another thing to note is that the graph of sin(x) has 180 degree point symmetry about the origin, which means that sin(x) is an odd function of x." [p. 35]

Neat. I did not realize this.

• The graph of cos(x) is similar to that of sin(x) except that it begins at +1 instead of 0. [p. 36]
  
  
• "Furthermore, notice that this time the graph has mirror symmetry in the y-axis. This means that cos(x) is an even function." [p. 36]

I did not realize this either. I need to develop a reflex for looking for symmetry when I view graphs.

• The graph of tan(x) has vertical asymptotes at p/2 and -p/2. It is symmetic about the origin and is therefore an odd function of x.

I now have these 3 graphs under control. But Banner goes on to describe the graphs for the next 3 trigonometry functions: secant, cosecant and cotangent.

• Given the graph of sin(x) it is easy to see that the graph of sec(x) = 1/sin(x) will have a vertical asymptote whenever sin(x) is zero (i.e. at multiples of p/2). A glance at the graph also shows that sec(x) is symmetric about the y-axis and is therefore an even function. [p. 37]

Let me think about this. Consider any function that is odd. Now consider its reciprocal. It must be even. I am delighted with this insight! I thought of this before I read it.

• Now consider the graph of csc(x) = 1/cos(x). This will have vertical asymptotes at multiples of p. Also cos(x) is an even function. A look at the csc(x) graph shows it to be an odd function.
  
  
• Similar comments apply to the cot(x) function.

Oops. The graph of tan(x) is odd. So is the graph of cot(x) = 1/tan(x). My earlier insight was wrong. The moral here is to not only look for patterns but to check them out to see if they are really general.

 

I have completed the section on graphing trig functions. But I have not actually graphed any of the functions. There is no handwritten work this morning. The next section covers many of the basic trig identities. I also want to begin looking at Mathematica. Total time this morning: about 1 hour.

Tags: mathematics, calculus