Trig Functions in Three Problems: Graphing, Solving, and Factoring

Date: 05/15/98 at 04:53:07
From: Carolyn
Subject: graph sin x - cos x

I am taking Math 12 by correspondence, and it's been years since I was 
in school. Since I haven't had math since grade 10, I'm having a few 
problems. I actually have three questions:

   1) Sketch the graph of sin(x) - cos(x)  
      Have I figured this out correctly? 
      sqrt(1^2 + (-1)^2) = sqrt(2)
      cos(x) = 1/sqrt(2), sin(x) = -1/sqrt(2), x = - pi/4

   2) Solve: sqrt(3)*cos(x) = sin(x)       
      I don't even know where to start with this one.  

   3) Factor sin(x)cos^2(x) - 9 sin(x) - 9 cos(x) + sin^2(x)cos(x).
      I'm having a real problem with factoring. I can't seem to grasp   
      the concept of it. It appears to me that when something is 
      factored, parts of the original equation are just deleted. 

I hope that you are able to help me.

Date: 05/15/98 at 08:21:17
From: Doctor Jerry
Subject: Re: graph sin x - cos x

Hi Carolyn,

On problem 1, it looks as if you started correctly, but something 
happened along the way. For a*sin(x) - b*cos(x), calculate 
sqrt(a^2 + b^2). In your case, this is sqrt(2). Multiply and divide 
by it:

   sqrt(2)[(1/sqrt(2))*sin(x) - (1/sqrt(2))*cos(x)]

Since cos(pi/4) = 1/sqrt(2) and sin(pi/4) = 1/sqrt(2), we can write:

   sqrt(2)[(1/sqrt(2))*sin(x) - (1/sqrt(2))*cos(x)]
      = sqrt(2)[cos(pi/4)*sin(x) - sin(pi/4)*cos(x)]
      = sqrt(2)[sin(x + pi/4)]

OK, this is much easier to graph.

In problem 2, you have sqrt(3)*cos(x) = sin(x). Well, you can just 
write:

   tan(x) = sqrt(3)

For one thing, x = pi/3 works.

In problem 3, you need to factor:

   sin(x)cos^2(x) - 9*sin(x) - 9*cos(x) + sin^2(x)cos(x)

Let your mind look for patterns (of course, you must first have some 
patterns stored).

From first and last terms:

   sin(x)*cos(x)*(cos(x) + sin(x))

From middle two:

   -9(sin(x) + cos(x)).

So:

   sin(x)*cos^2(x) - 9*sin(x) - 9*cos(x) + sin^2(x)*cos(x)
        = sin(x)*cos(x)(cos(x) + sin(x)) - 9(sin(x) + cos(x))
        = (sin(x) + cos(x))(sin(x)*cos(x) - 9)

-Doctor Jerry, The Math Forum
Check out our web site! http://mathforum.org/dr.math/