Expert: Scotto - 3/24/2009

Question
Hi! I have come across some problems in math for a placement test for college and i have no idea what to do with them! I have no idea how to do the following problems so any help and explanations would be great.

1. if cos x= a/3 and 0<x<pi/2 then what is tan x?
2. what are the solutions of sin^2 x +cos x=1 with 0 less than or equal to x < 2pi
3. simplify the trigonometric expression
(cos^2 x)(tan x + cot x)


Answer
1. If the cos(x) = a/3, the hypotenuse of the triangle has length a and the near side has length 3.  
This means that the far side has length sqrt(9-a*a).  Since the angle is in the first quadrant, that side length is positive.  This means that the tangent of that angle is sqrt(9-a*a)/3.


2. We know that sin^2(x) is 1-cos^2(x), the equation is solving
1-cos^2(x) + cos(x) = 1.  The 1’s cancel out,
leaving -cos^2(x) + cos(x) = 0.  

This factors into cos(x)(1 – cos(x)) = 0.  

The solutions are where cos(x) = 0 or cos(x) = 1.

From what is know of the way the cos() curve looks,
the first equation has the solution x=pi/2 and x=3pi/2.
The second equation has the solution x=0.


3. cos^2(x)(tan(x) + cot(x)) =
cos^2(x)(sin(x)/cos(x) + cos(x)/sin(x)) =
cos(x)sin(x) + cos^3(x)/sin(x).

Or you could say that tan(x) + cot(x) is
sin(x)/cos(x) + cos(x)/sin(x)
= sin^2(x)/(sin(x)cos(x)) + cos^2(x)/(sin(x)cos(x))
= (sin^2(x) + cos^2(x))/(sin(x)cos(x))
= 1/(sin(x)cos(x)),
so if this is mutliplied by cos^2(x), you get
cos^2(x)/(sin(x)cos(x)) = cos(x)/sin(x) = cot(x).