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what are the possible values of x in 2sin^2(x-30)=cos 60
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They want you find all real solutions to this equation
2sin^2(x-30) = cos(60). (1)
We have
cos(60) = 1/2; so, this equation takes the form
2sin^2(x-30) = 1/2.
Simplify it step by step
sin^2(x-30) = 1/4,
sin(x-30) = +/-},
sin(x-30) = +/- 1/2. (2)
From this equation, there are two infinite sets of roots for x-30 to equation (2)
x-30 = , k = 0, +/-1, +/-2, . . .
x-30 = , k = 0, +/-1, +/-2, . . .
It gives two infinite sets of roots for x to equation (1)
x = , k = 0, +/-1, +/-2, . . .
x = , k = 0, +/-1, +/-2, . . .
ANSWER. Equation (1) has two infinite sets of solutions
x = , k = 0, +/-1, +/-2, . . .
x = , k = 0, +/-1, +/-2, . . .
Solved.
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To tutor @greenestamps and to all other readers of this post.
Thank you for your attention, but my solution, as it is/was written by me
from the very beginning, is correct and includes all the cases sin(x-30) = +/- 1/2.
It is because the general formula uses not full period , but half period , which absorbs/covers all these cases.
So, everything is and was correct in my post. There is no reason to worry.
This alarm from @greenestamps is false.