SOLUTION: Please help! I have already done most of the work. Recently I asked about finding all real numbers in the interval [0, 2pi] that satisfy the equation. I have worked through th

Question 1004805: Please help! I have already done most of the work.
Recently I asked about finding all real numbers in the interval [0, 2pi] that satisfy the equation.
I have worked through the problem but I feel like I have gotten stuck in some places. Here's the problem and where I'm at. Thank you in advance for your help!

=
= and .
Starting first with .
=
=
Now, I've seen other examples that use and have the results be 0, as well as and ...but I'm not quite sure why those last two should or shouldn't be included...why not or ? If you divide their sin by their cosine, doesn't it equal zero? So why choose and with the zero and not others? Should I just leave it at 0, or should I include and ?
Moving on to .
=
=
=
=
=
But now where do I go? Not quite sure how to find the arcsecant of something with a square root...How do I type that into my calculator? (I dont have a graphing calculator...but I can do sines, cosines and tangents with their inverses).
Help would be greatly appreciated. Thank you! :)
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
.

1.  Regarding first part of your question.

You got 

tan(x) = 0. 

The solutions of this equation are  x = 0, +/-, +/-, +/- , . . . , +/-, . . . .

Of them,  only 0,  and   belong to the segment  [, ]  that you pointed.

Other values that you are asking about,  like , ,  are not the roots of the equation  tan(x) = 0.


2.  Moving on to   -  = .

It is   = +/-,     or      = +/-     or     cos(x) = +/-.

If  cos(x) = ,  then x =  and x =  =  are the solutions in the segment  [, ].

If  cos(x) = -,  then x =  and x =  =  are the solutions in the segment  [, ].


I recommend you to make a sketch of the unit circle, to mark the angles 0�, 30�, 45�, 60� and 90� in it and to write 

the values of sin, cos and tan for these angles.

Then mark all other angles like 120�, 135�, 150�, 180� and so on till 360� and do the same.

After that you will be much more confident in such problems.

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