Easy trig. (#2) (1 Viewer)

[sinophile]

Man I can't seem to advance a single question without hitting obstacles.. Sorry for the second topic (putting it in the old one won't receive any hits)

Solve for 0<=x<=360:

sin2x=1/2.

My working out:
Let 2x be 'u'.
sinu=1/2.
Since Sin2x is positive, u lies in lst and 2nd quadrant.
Therefore, u=30, 150.
Hence 2x=30, 150
x=15, 75.

But wait! The textbook solution:
x=15, 75, 195, 255

Can someone please explain why this is? How come there are more values for x? How do we get these values? Thanks!

 

[Timothy.Siu]

Man I can't seem to advance a single question without hitting obstacles.. Sorry for the second topic (putting it in the old one won't receive any hits)

Solve for 0<=x<=360:

sin2x=1/2.

My working out:
Let 2x be 'u'.
sinu=1/2.
Since Sin2x is positive, u lies in lst and 2nd quadrant.
Therefore, u=30, 150.
Hence 2x=30, 150
x=15, 75.

But wait! The textbook solution:
x=15, 75, 195, 255

Can someone please explain why this is? How come there are more values for x? How do we get these values? Thanks!

its because 0<=2x<=720
u dont need substitution, its wasting a step.
u shud just do sin 2x=1/2
2x=30 or 150 or 390 or 510
x=15 or 75 or 195 or 255

 

[kwabon]

sin 2x = .5 0<x<360

The sin 2x means revolutions
obviously you are used to these type of question sin x = .5 this means that you only have to take one revolution into consideration
so therefore the 2x means that you have to take 2 revolutions into consideration.
so hopefully you know what the 2x means

sin 2x = .5
2x = sin ^-1 (.5)
2x = 30 , 150 (sine is positive in two quadrants)
2x = 30 , 150 , (30 + 360) , (150 + 360) (remember how i said that u have to take 2 revolutions into consideration, so u add each answer by 360)
2x = 30 , 150 , 390 , 510
x = 15 , 75 , 195 , 255 (divide by 2)

and there you go
hope you understood, if anyone can explain it better, go for it!

 

[sinophile]

Huh? I dont get your explanation for the two revolutions.

What i'm reading your explanation as, since its sin2x, then the domain is 0<=2x<=720? But im not getting the solution for 2x, im getting the solution of x, so shouldn't the domain just be 0<=x<=360?

 

[Timothy.Siu]

Huh? I dont get your explanation for the two revolutions.

What i'm reading your explanation as, since its sin2x, then the domain is 0<=2x<=720? But im not getting the solution for 2x, im getting the solution of x, so shouldn't the domain just be 0<=x<=360?

but when u are solving sin 2x=0.5, u are solving it for 2x
hence u have to double the domain,i.e. 0<=2x<=720

 

[sinophile]

but when u are solving sin 2x=0.5, u are solving it for 2x
hence u have to double the domain,i.e. 0<=2x<=720

But then when you solve the equation sin2x=0.5 for x, then you keep the domain the same.. Right? Or am I missing something?

 

[clintmyster]

But then when you solve the equation sin2x=0.5 for x, then you keep the domain the same.. Right? Or am I missing something?

no the domain doubles as timothy said.

Remember the question is asking to Solve for 0<=x<=360:

but when finding without substitution, were finding 2x and therefore have to adjust our original domain by 2. Therefore its 0<=2x<=720

 

[sinophile]

no the domain doubles as timothy said.

Remember the question is asking to Solve for 0<=x<=360:

but when finding without substitution, were finding 2x and therefore have to adjust our original domain by 2. Therefore its 0<=2x<=720

I follow your train of logic, but I don't know the underlying reasons why it is.. In other words, i know how, i dont know why. Can you please explain why we need to double the domain, and why there are so many values of x? Thanks.

 

[kwabon]

just draw it up
i am pretty sure u noe how to draw sin 2x

so just draw up sin 2x and see where it equals .5, u will c there are 4 answers.
and thats why u need to double the domain and then divide by 2

 

[annabackwards]

I follow your train of logic, but I don't know the underlying reasons why it is.. In other words, i know how, i dont know why. Can you please explain why we need to double the domain, and why there are so many values of x? Thanks.

The domain given is for X, ie 0<=x<=360

But you 1st need to work with 2x as this is the degree given for sin, so the domain doubles, ie 0<=x<=720

Then after you solve for 2x you need x which is what their asking for, so you divide your solutions for 2x by 2. Remember to include both the solutions for 2x and x in your answer of you'll get it wrong!!!

By the way, I worked out the question for you on paper because i can't stand typing up math on the comp- with full working out because it might help you.
You can find it here (excuse my horrid handwriting): http://i238.photobucket.com/albums/ff62/annabackwards/math.jpg

 

[sinophile]

The domain given is for X, ie 0<=x<=360

But you 1st need to work with 2x as this is the degree given for sin, so the domain doubles, ie 0<=x<=720

Then after you solve for 2x you need x which is what their asking for, so you divide your solutions for 2x by 2. Remember to include both the solutions for 2x and x in your answer of you'll get it wrong

By the way, I worked out the question for you on paper because i can't stand typing up math on the comp- with full working out because it might help you.
You can find it here (excuse my horrid handwriting): http://i238.photobucket.com/albums/ff62/annabackwards/math.jpg

Thank you, although I still don't get the underlying reason..

 

[annabackwards]

Thank you, although I still don't get the underlying reason..

I dunno how else to explain it... you basically find 2x because that's all you can do and then you use normal algebra to find x (ie you divide both sides by 2)

 

[Timothy.Siu]

well, if u draw it, u can see it has 4 solutions from 0<=x<=360
basically because its sin 2x, the period (wavelength) is halved, so basically u know the solution is 15 and 75 but each 180 degrees it repeats itself....so 195 and 255 also fit the domain...

 

[kurt.physics]

Thank you, although I still don't get the underlying reason..

I'll try help.

this question starts out with; solve for 0 < x < 360.

So lets say we have to solve sin (x) = 1/2. We would see that using the equilateral triangle that x = 30 degrees. But as we know there are more solutions, for example if this sine was taken from a triangle, in the second quadrant, then the sin would be positive, so there is another solution x = 180 - 60 = 120 degrees.

So the aim for these kind of questions is to find what x angle make it a 1/2 or what ever.

But what about 2x, well, for sin (2x) = 1/2. First note that 2x is the angle, so dont consider for a while that its 2 times x, but 2x is one angle (lets call it A), say 30 degrees or 72 degrees (not in this example of course)

Now the question is for x thats between 0 and 360 degrees ie 0 < x < 360. But we have 2x, so when we look for solutions, we have to look at all the angles from 0 to 720 ie 0 < 2x < 720. The reason is that we multiply but 2, ie 0 X 2 < 2 times x < 2 X 360. So we will find solutions for 2x or A if you will, remember that we are considering 2x as just one thing for now, not just 2 times x.

But when we find that 2x is 30, 150, 390, 510 we have solved for this number A (aka 2x). Now this is where we consider this 2x to not be A (A is just any number) or not one whole this, we now consider it to be 2 times x. So if 2x = 30, then dividing by 2 one both sides we get x = 15 and so on.

So

2x = 30, 150, 390, 510

x = 15, 75, 195, 255

But you might wonder, when we divide by 2x, we divide 0 < 2x < 720, getting back to 0 < x < 360.

What the other people mentioned about cycles and revolutions is the more complicated way of seeing this problem.

Hope that clears everything up!

 

[sinophile]

Thank you so much for your help everoyne!

 

[annabackwards]

Thank you so much for your help everoyne!

Am i right in assuming you understand now?

If so, congrats!!!