I have to solve the following problems, and frankly I don't know how.  So if anyone could solve them for me, and possibly show work & explain the process, it'd be appreciated:

(I will indicate the unit circle [theta] as (-))

1.) tan x = 2.5 (0 <= x < 2pi)
2.) 3cos x + x = sin x (0 <= x < 2pi)
3.) (tan^2 x - 1) * (cos^2 x - 1) = 0 (0 <= x < 360)
4.) sin A = 2/3, cos B = 7/8, sin(A + B) = ? (pi/2 <= A <= pi && pi/2 <= B < 0)
5.) cos ((-) + pi/2) = -sin(-) (prove the identiy)
6.) y = sin^-1 x (Domain:{...} Range:{...})
7.) csc^-1 5 = ?
8.) 1 - tan^2 (-) / 1 + tan^2 x = cos^2 x (prove the identity)
9.) 2sin^2(-) - sin(-) - 1 = 0 (solve with quadratic formula)

How much time do you have to do these?  Better to think about them than have the solutions handed on a silver platter :P.

Of course 1) is rather easy: take the inverse of both sides; 
arctan(tanx) = arctan(2.5) ---> x = arctan(2.5)  |    0 <x<2pi.

3) can be solved by looking up the identity for sin(x+y), or proving the identity
by using Euler's theorem,  e^(ix) = cos(x) + i*sin(x).  --> sin(x) = [e^(ix)-e^(-ix)]/2i  ---->  you do the rest :P.
I don't see how B could be greater than or equal to pi/2 and less than zero.

5) can be done graphically or by using the formula used in 3  (it doesn't really matter if you're taking sine or cos on the unit circle or not for this one).

7)  Hmm, you must be allowed a calculator

9)  You are told how to do it. 


6) is interesting.  arcsin is actually a multivalued function and often a
single valued branch is chosen for it.  your teacher should clarify this

let me know how you do 2)

What are you allowed? I'm supposing you haven't talked about taylor series
or anything?  Allowed a graphing calculator? 


Is there a typo in 8?  Why are you dividing by 1?

[quote author=Rule link=topic=14653.msg149573#msg149573 date=1143939192]
How much time do you have to do these?  Better to think about them than have the solutions handed on a silver platter :P.

Of course 1) is rather easy: take the inverse of both sides; 
arctan(tanx) = arctan(2.5) ---> x = arctan(2.5)  |    0 <x<2pi.

3) can be solved by looking up the identity for sin(x+y), or proving the identity
by using Euler's theorem,  e^(ix) = cos(x) + i*sin(x).   --> sin(x) = [e^(ix)-e^(-ix)]/2i  ---->  you do the rest :P.
I don't see how B could be greater than or equal to pi/2 and less than zero.

5) can be done graphically or by using the formula used in 3  (it doesn't really matter if you're taking sine or cos on the unit circle or not for this one).

7)  Hmm, you must be allowed a calculator

9)  You are told how to do it. 


6) is interesting.  arcsin is actually a multivalued function and often a
single valued branch is chosen for it.  your teacher should clarify this

let me know how you do 2)

What are you allowed? I'm supposing you haven't talked about taylor series
or anything?  Allowed a graphing calculator? 


Is there a typo in 8?  Why are you dividing by 1?
[/quote]

Hey, thanks for helping out.  Well, to tell you the truth, I got really behind on this part of Trigonometry and am trying to catch up quickly, but this was obviously a difficult & critical unit (analytical trigonometry) & these are actually the questions from my test, I wrote them down (cheating) & am trying to solve them at home.

Really though, I want to learn this stuff, and not just get the answers (which is why I wanted them explained).

Addressing your response:

- I have until Sunday evening to complete these.
- For #1, I assumed taking the inverse so it would be x = tan^-1 2.5.  However, I get a decimal, and other students have a set of radians for their solutions.
- For #3, we haven't covered Euler's theorom.  I don't quite understand what you're saying here either.
- For #5, how would I solve this graphically?
- For #7, I plugged this into a calculator and got an undefined value.
- For #9, I did this before posting this and got x = +- 3sin/4sin.  I'll try again after this post.
- For #2, I'll start doing that one after this post.
- I'm allowed a graphing calculator and a list of the basic trigonometric identities (recipricol, even-odd, pythagrean) and sum&difference, double-angle, and another set of identities I can't remember.
- There's no typo in #8, I just forgot to indicate that it's being divided by the expression, forgot to put parentheses.

Hahah.  Was just reading through some old posts to see what kind of math questions people had back in the day, and came across this.  Amazing to see what some of us once thought was difficult ;)