Question

Solve. Finding all solutions in (0,2π)
cos(π/2 +x) + sin(x-π)=1

Answer 1

\[\cos (\frac{\pi}{2}+x)+\sin (x- \pi)=1.\]\[\implies - \sin x + [\sin- (\pi -x)]=1.\]\[\implies - \sin x- \sin (\pi -x)=1.\]\[\implies - \sin x -\sin x=1 \implies -2 \sin x=1 \implies \sin x=-\frac{1}{2}.\]

Answer 2

Now find 'x' in accordance with the interval \[\LARGE{\color{RED}{(0, 2 \pi).}}\]

Answer 3

unfortunately i only have four anser choices i forgot to mention and they are either:
a- 7π/6,11π/6
b- 7π/6
c-π/6, 7π/6
d- π/4, 3π/4

Answer 4

I tell you how to do it just wait..

Answer 5

just figure out the answer by last equation which we got
\[\implies \sin x=-\frac{1}{2}.\]

Answer 6

It would be option A

Answer 7

don't u get
then
wait for @waterineyes :D

Answer 8

how did you get it from the original equation to −sinx+[sin−(π−x)]=1.

Answer 9

is there a certain rule/identity you follow

Answer 10

I have written the solution above
didn't u see it????????/

Answer 11

ya there are many identities
but those which i used are as follows
\[[1.] \space \space \cos (\frac{\pi}{2}+x)=- \sin x.\]\[[2.] \space \space \sin -( x) = - \sin x.\]\[[3.] \space \space \sin (\pi- x)=\sin x.\]

Answer 12

ok, now i get how you got to it. For the first one it looks familiar, can you tell me what its called please.

Answer 13

the rule/identity

Answer 14

Reduction formulae.

Answer 15

thanks

Answer 16

welcome:D
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Symmetry.2C_shifts.2C_and_periodicity

Answer 17

\(\cos (\frac{\pi}{2}+x)+\sin (x- \pi)=1\)

\(\implies - \sin x + (\sin- (\pi -x))=1\)

\(\implies - \sin x- \sin (\pi -x)=1\)

\(\implies - \sin x -\sin x=1 \implies -2 \sin x=1 \implies \sin x=-\frac{1}{2}.\)

\[sinx = -\sin (\frac{\pi}{6}) \implies sinx = \sin(\frac{- \pi}{6})\]

Here the general solution is given by:

\[x = n \pi + (-1)^n (\frac{- \pi}{6})\]

Put the values of n as 0 , 1 , 2 etc etc..

Answer 18

Put n = 0:

\[x = \frac{- \pi}{6} \implies \frac{11 \pi}{6}\]

Answer 19

Put n = 1:

\[x = \pi + \frac{\pi}{6} \implies x = \frac{7 \pi}{6}\]