Question

Solve: sinx+cosx+sin2x>1

Answer 1

i imagine it gon be a hell of a lengthy work :D

Answer 2

I know that sin2x=2sinxcosx, but i cannot facotorize this on right way :(

Answer 3

just graph it. Save tones of times :DD

Answer 4

Yeah and wolfram saves time, but i need procedure :)

Answer 5

yeah this wont factor at all
you just have to recognize that at x=0 you get 1 and at x=pi/2 you get 1
so its greater than 1 in the interval (0,pi/2)

Answer 6

and how i know that function f:R->R f(x)=sinx+cosx+sin2x doesnt intersect x ray between this segment?

Answer 7

sin(x) + cos(x) + sin(2x) >1
sin(x) + cos(x) + 2sin(x)cos(x) >1
Replace the 1 on the right with sin^2(x) + cos^2(x)
sin(x) + cos(x) + 2sin(x)cos(x) > sin^2(x) + cos^2(x)
sin(x) + cos(x) > sin^2(x) + cos^2(x) - 2sin(x)cos(x)
sin(x) + cos(x) > {(sin(x) - cos(x)}^2
The RHS being a square is always >= 0
So we need to solve for sin(x) + cos(x) > 0

Answer 8

Maybe the conclusion in the last line is not correct.

Answer 9

@ranga I was gonna say so :DD

Answer 10

But if I plot sin(x) + cos(x) + sin(2x); y = 1 and sin(x) + cos(x)
the three graphs nicely intersect at all the points where the original function is > 1.

Answer 11

:ranga it isnt correct beacause for 2pi/3 sin(x) + cos(x) > 0, but sin(x) + cos(x) + sin(2x) >1 is not correct...

Answer 12

well, consider x > x^2. The solution isn't x > 0 simply because x^2 > 0

Answer 13

0 <= ( sin(x) - cos(x) )^2 <= 2
can this fact be used?

Answer 14

idk :D why is that needed? :D

Answer 15

because sin(x) + cos(x) > {(sin(x) - cos(x)}^2

Answer 16

Is that sin^2x or sin(2x)?

Answer 17

sin(2x)

Answer 18

Wherever sin(x) + cos(x) > 1 so is the original function.

Answer 19

how i know that? :D

Answer 20

without looking graph...

Answer 21

I am reading from the graph and trying to see if that fact can be used as a hint.

Answer 22

your solution indeed matches with the solution provided by wolfram alpha but I'm still trying to figure out how 0 <= ( sin(x) - cos(x) )^2 <= 2 is good fact to use

Answer 23

i got it... substitution x=pi/4-y solves it :)