Question 550923
For what numbers <font face = "symbol">q</font>, 0&#8806;<font face = "symbol">q</font>&#8806;2<font face = "symbol">p</font> does cos<font face = "symbol">q</font> = 1? 
Can you please show me a step by step instruction on how you solve this problem.
<pre>
On the unit circle, each point is (cos(<font face = "symbol">q</font>),sin(<font face = "symbol">q</font>)), where <font face = "symbol">q</font> is the angle between 
the right hand side of the x-axis and the radius drawn to that
point:

{{{drawing(400,400,-2.7,2.7,-2.7,2.7, circle(0,0,1), graph(400,400,-2.7,2.7,-2.7,2.7),circle(cos(.6),sin(.6),.05),
line(0,0,cos(.6),sin(.6)),locate(.33,.23,theta), locate(.9,.7,P(cos(theta),sin(theta)))

  )}}}

So you want to know where cos<font face = "symbol">q</font> = 1.

Since the x-coordinate of every point on the unit circle IS the
cosine of the angle that the radius to that point makes with
the right side of the x-axis, we are looking for a point on that
circle with an x-coordinate of 1.  There is only one point on
the unit circle that has its x-coordinate as 1, and that is the
point (1,0)

{{{drawing(400,400,-2.7,2.7,-2.7,2.7, circle(0,0,1), graph(400,400,-2.7,2.7,-2.7,2.7),  locate(1.1,.22,P(cos(0),sin(0))),line(0,0,1,0),circle(1,0,.05)

  )}}}{{{drawing(400,400,-2.7,2.7,-2.7,2.7, circle(0,0,1), graph(400,400,-2.7,2.7,-2.7,2.7),  locate(1.1,.22,"(1,0)"),line(0,0,1,0),circle(1,0,.05)

  )}}}

and the value of <font face = "symbol">q</font> there is either 0 or 2<font face = "symbol">p</font>.  

So that's the answer.  <font face = "symbol">q</font> = 0 or 2<font face = "symbol">p</font>.
 
Edwin</pre>