Radian Angle Solving: 0-2π

  • Thread starter thomas49th
  • Start date
  • Tags
In summary, the conversation discusses solving the equation 2sin(x-\frac{\pi}{3}) = 1 for the range 0 <= x <= 2π. The first solution is π/2 and the second solution is found by adding π/3 to both values of π/6 and 5π/6, resulting in the solutions π and 5π/6. The conversation also mentions that the solutions for x must be within 2π.
  • #1

Homework Statement

[tex]2sin(x-\frac{\pi}{3}) = 1[/tex]
for the range 0 <= x<= 2π

The attempt at a solution

so [tex]x-\frac{\pi}{3} = \frac{\pi}{6}[/tex]

i can get the first solution which is π/2 but how do i get the next solutions?

Thank you
Physics news on Phys.org
  • #2
Examine the graph of a sine function between 0 and 2π. Find all angles where sine = 1/2. π/6 is only one of them.
  • #3
well there is 1 more in the range of 0 to π to 360

that is π - π/6
= 5π/6

so the answer is

π and 5π/6

is there any more? i presume i don't count the -1/2s

thanks :)
  • #4
thomas49th said:
well there is 1 more in the range of 0 to π to 360

that is π - π/6
= 5π/6
Right. So the two values of [itex]\theta[/itex] that satisfy [itex]\sin\theta = 1/2[/itex] are [itex]\pi/6[/itex] & [itex]5\pi/6[/itex].

Given that, what are the values for x?
  • #5
do you add π/3 to both of them?

  • #6
  • #7
This is the way I was taught to do it :
[tex]2\sin (x-\frac {\pi}{3})= 1[/tex]
[tex]\sin (x-\frac {\pi}{3})= \frac {1} {2}[/tex]
Let [tex]q =x-\frac {\pi}{3} [/tex]
[tex]\sin (q)= \frac {1} {2}[/tex]
Where is the [tex]\sin q = \frac {1}{2}[/tex] ?
At [tex]\frac {\pi}{6}, \frac {5\pi }{6}[/tex]
Thus : [tex]q_{1} =\frac {\pi}{6} , q_{2}=\frac {5\pi }{6}[/tex]
We're not done. We still have to solve the x. Note that I was supposed to add 2 Pi to q 1 and q 2, but if you do it separately, you will see the solutions would not be needed since they are outside of 2 Pi when we add pi /3.
Continuing : Simply setting the q's equal to x - pi /3
[tex]x-\frac {\pi}{3} =\frac {\pi}{6}[/tex]
[tex]x-\frac {\pi}{3} =\frac {5\pi }{6}[/tex]
Solving, we get the solutions to be : [tex]x_{1} = \frac {\pi}{2},x_{2} = \frac {7\pi}{6},[/tex]

In your calculator, if you graph these two functions, you will see the solutions to be those as noted.
  • #8
Thanks! :)

Suggested for: Radian Angle Solving: 0-2π