Precalculus question using trigonometric equations

[grafs50]

Homework Statement

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

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

 

[Infrared]

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

 

[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)

 

[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.

 

[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).

 

[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.

 

[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.

 

[grafs50]

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

 

[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

 

[Rimi]

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

 

[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.

 

[HallsofIvy]

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.

 

[Rimi]

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.

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

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

 

[grafs50]

Alright thanks everybody.