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.
that.... is alot of equation manipulation...
to avoid losing answers, first step would be to move the equation on the right to the left, so it will be
(1+sin x)/cos x + cos x/(1+sin x) - 8 sin^2 x cos x.=0
next will be getting rid of the demominator...
a bit too long to solve offhand...
Originally posted by bonkysleuth: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.
wah !
Note that you should be in radian mode for this question while checking.