Find all the values of x between -3 and 3 which satisfy the
equation:
(1+sin x)/cos x + cos x/(1+sin x) = 8 sin^2 x cos x.
Express your answer in terms of pi.
[(1+sin x)^2 + (cos x)^2]/[(1+sin x)(cos x)]= 8 sin^2 x cos x
(1 + 2 sinx + sin^2 x + cos^2 x)/[(1+sin x)(cos x)] = 8 sin^2 x
cos x
2(1+ sin x)/[(1+sinx)cos x] = 8 sin^2 x cos x
(1+sin x ) = 4(1+ sin x) sin^2 x cos ^x
(1+ sin x) (1- 4 sin^2 x cos ^2 x) = 0
(1+ sin x)( 1- sin^2 2x)= 0
sin x= -1 or sin 2x = -ve/+ve 1
From sin x = -1
x = -pi/2 (NA) as the denominator will be zero
From sin 2x= +ve/-ve 1
2x = -3pi/2, -pi/2, pi/2, 3pi/2
x= -3pi/4, -pi/4, pi/4 or 3pi/4
Note: made a mistake in mental calculation earlier as I was
going out and didn't check my workings.
"4sin^2 x cos^2 x" is "sin^2 2x", not "2sin^2 2x" as posted
previously. Solution adjusted. Sorry for any confusion.
To make up for it, I will reveal an effective method on how to
check whether your solutions are correct within 2mins.
1st you need a calculator with playback and saved "Ans"
Best is Casio FX 95SG for easy keying in of orginal
relationship
1. Input the 1st solution "-3pi/4" and press "="
Now "-3pi/4" is saved as the "Ans".
2. Key in the original equation with "Ans" as the
replacement for x with the terms on the RHS brought to the
LHS.
3. Press "=" ( solution is correct if you see 0)
4. Now press "-pi/4" and "="
5. Scroll "replay" until the the original equation is on the
screen and press "="
6. Now press "pi/4" and "="
7. Scroll "replay" until... Repeat as necessary.
Is the Casio FX 95 SG worth the $20? Let's just say I bought 20
of them.
There are plenty of other ways in which a good calculator can
help you score better in the O level.