Expert: Jack Cheng - 1/1/2007

Question
SOLVE for X in each of the following:
a) X is between 0 and 2pi (meaning show answer in radians??)
2sin^2 x = 2 + cos2x

I substituted using a double angle formula: cos2A=1-2sin^2 A.
cos 2x=2sin^2 x -2
1-2sin^2 x=2sinx-2
-2sin^2 x=2sinx-3
2sin^2 x +2sinx-3=0
Let y= sinx
2y^2 + 2y-3=0
I was unable to factor this and I used the quadratic formula. I ended up with x= -1 plus/minus square root of 7 divided by 2. is this anywhere close? If so what do I do next?


b) X is between 0 degrees and 360 degrees (meaning show answer in degrees??)
sin2x + cosx = 0

sin2x=-cosx
2sinx cosx= -cosx
2sinx + cos2x=0
2sinx + (1-2 sin^2 x) =0
2sinx + 1-2 sin^2 x =0
----I divided by two on both sides-----
sinx - sin^2 x = 0

Did I do anything wrong? If I did it right then what do I do next?


c) X is between 0 degrees and 360 degrees (meaning show answer in degrees??)
3sin(x + 20) degrees = -1

3sin(x + 20)= -1
---I divided both sides by 3---
sin(x+20)= -1/3
Now what? I think that I need to find a reference angle or something like that...

Please help me!!

I understand that this is quite a lenghty question but any help you can offer would be GREATLY appreciated. Thank you in advance for your time.  

Answer
Hi Jon,

a) you're probably right in that it's looking for an answer in radians.

Your answer is a bit off though. In about step 2, you forgot the square on the sin x. It should be 1-2sin^2(x) = 2sin^2(x)-2. You can then combine the sin^2 terms and eventually get +/- (pi/3) for the answer.

b) I am not sure what you did, especially starting with step three. But from step two, you can just divided by cos x and get 2sin(x)=-1. You can solve it easily from there.

c) I am not sure if this is an all mental-math problem, or if you're allowed to use a calculator. If you can use a calculator, then just use the inverse sine function.  The answer is what you get with the sine inverse function minus 20.

I hope this helps,
Jack