SOLUTION: Solve the following equation on the interval [0, 360 degrees]: sin2x=1 This is what I have done so far: Let 2x = theta sin theta = 1 theta = 90 degrees 2x = 90 --> x = 4

Algebra ->  Trigonometry-basics -> SOLUTION: Solve the following equation on the interval [0, 360 degrees]: sin2x=1 This is what I have done so far: Let 2x = theta sin theta = 1 theta = 90 degrees 2x = 90 --> x = 4      Log On


   

Question 1103336: Solve the following equation on the interval [0, 360 degrees]:
sin2x=1
This is what I have done so far:
Let 2x = theta
sin theta = 1
theta = 90 degrees
2x = 90 --> x = 45 degrees
Add period to theta:
90 + 180 = 270 --> x = 270/2 = 135 degrees
270+ 90 = 450 --> x = 450/2 = 225 degrees
I don't understand why the answer in my book says "x = 45 degrees, 225 degrees". According to my work done, shouldn't x = 135 degrees as well?
Help with this question would be really appreciated!
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
sin(2x) = 1  ====>

2x = pi%2F2.


The last equation 2x = pi%2F2 has TWO solutions in the interval [0,2pi).


One of these two solutions is  x%5B1%5D = pi%2F4,  which is OBVIOUS.


The second solution  is  to add  pi  to the first solution:

    x%5B2%5D = pi%2F4+%2B+pi = 5pi%2F4.


Why ?  - Because when you take  2%2Ax%5B2%5D  with  x%5B2%5D = pi%2F4+%2B+pi,

you will get  2%2Ax%5B2%5D = pi%2F2+%2B+2pi,


which is GEOMETRICALLY THE SAME AS the ANGLE pi%2F2.


So your  2%2Ax%5B2%5D  is geometrically the same  pi%2F2  and satisfies the original equation  sin(2x) = 1.


Notice that both  x%5B1%5D = pi%2F4  and  x%5B2%5D = pi%2F4+%2B+pi belong to the interval [0,2pi), so they both are the solutions 

to your original equation  sin(2x) = 1  in this interval.



The plot below confirms my arguments:





Plot y = sin(2x)  (red)  and  y = 1 (green)


------------------
Do you understand this explanation ?

If "NO", send a message to me through the "Thank you" window/form.

If "YES", send a message to me through the "Thank you" window/form, and then I will explain more to you related to this kind of equations.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Well, for one thing, 135 degrees does not satisfy the equation. sin%282%2A135%29+=+sin%28270%29+=+-1....

The period of sine is 360 degrees, not 180 degrees.

After you get your first answer of 2x=90, giving you the answer x=45 degrees, your next answer comes from 2x = 90+360 = 450, giving you the answer x=225 degrees.

I suspect that your confusion comes from thinking that, since the equation involves sin(2x), you only need to add 360/2=180 degrees to your first answer for 2x.

I always found it easier to avoid this kind of confusion by first solving the problem all the way through for the values of 2x and then getting my answers for x from those answers.

In this problem, we are to find the solutions to sin%282x%29=1%7D%7D+on+the+interval+%7B%7B%7B0%3C=x%3C360. Since our function is sin(2x), the prescribed interval 0%3C=x%3C360 becomes 0%3C=2x%3C720.

So now find all the solutions between 0 and 720 degrees where the value of sine is 1; they are at 90 degrees and 450 degrees. Then, since the values of 2x are 90 and 450 degrees, the values for x (the solutions we are to find) are 45 and 225 degrees.