# Precalculus question using trigonometric equations

1. Apr 12, 2013

### grafs50

1. The problem statement, all variables and given/known data
Can you all help?
The problem is 2 sin(x/2) - 1 = 0 I don't know what to do with the (x/2).

3. The attempt at a solution
So far I've done this:
2 sin(x/2)-1=0
2 sin(x/2)=1
sin(x/2)=1/2

2. Apr 12, 2013

### Infrared

Well if the sine of some angle is 1/2, what can you say about that angle?

3. Apr 13, 2013

### Hectorreturns

Now,

sin(x/2) = (1/2)
(x/2) = sin-1 (1/2)
(x/2) = (π/6) (Taking only principal value of sin)
x = (π/3)

4. Apr 13, 2013

### Infrared

Well this is the right idea, but it doesn't for just the principal value. Looking for values just in [0,2π), we get that x/2= π/6 or that x/2=5π/6 so x=π/3 or 5π/3. To get any other value of x, just add one of these values by a multiple of 2π.

5. Apr 13, 2013

### grafs50

Alright. But I need the answer in degrees.

So, I would just convert radians to degrees up there?
And I only understand how to come up with one of those two answers still (plug sin^-1(1/2) into a calculator; which gives degrees).

6. Apr 13, 2013

### Infrared

You should know how to convert radians to degrees. If you want to see how to get all values that give you a sine of 1/2, maybe try to draw a unit circle and see where y=1/2.

7. Apr 13, 2013

### grafs50

I was just making sure that that was in fact radians that I could convert to degrees. So I got x = 210, 330 degrees which is the same as the radian answers I saw above and work in the formula. Thanks for the help!

But I have another problem now... there is an equation (-1)+tan(3x/2)=0
I did this:
(-1)+tan(3x/2)=0
tan(3x/2)=1
then does tan x = 3/2?
x = tan^-1(3/2)? Because that doesn't seem to work.

8. Apr 13, 2013

### grafs50

Never mind, I just got it. x= 30, 210 degrees.

9. Apr 13, 2013

### grafs50

Actually how do I know how many solutions are in sin 2x= sqrt(2)/2?
I found x=22.5, but I don't have 22.5 degrees on the unit circle

10. Apr 13, 2013

### Rimi

Sine equals √2/2 when θ= 45°, 135°, etc. That should help

11. Apr 14, 2013

### grafs50

Thanks, I got 22.5 and 67.5 which seems to be right.

I just really don't understand how sin sqrt(2)/2 equals both 67.5 and 22.5 degrees. They are not the same on the unit circle. What about 112.5 and 157.5 degrees. And different rules seem to apply for tan and cos.

For Sine, does just the y-coordinates have to match? What about for Cos and Tan? We are working off-book so I can't look in the book to find how it works and I'm pretty confused.

Last edited: Apr 14, 2013
12. Apr 14, 2013

### HallsofIvy

Staff Emeritus
sine of a number does not equal any number of degrees! You seem to be very confused as to the basic definition, in terms of the unit circle, of the trig functions. You need to review that.

13. Apr 14, 2013

### Rimi

I agree with HallsofIvy. You need to review the unit circle.

Sin(2(22.5 degrees))= (2^.5)/2

14. Apr 20, 2013

### grafs50

Alright thanks everybody.