Plus Minus

Geometry Level 4

If we are given that $(1 + \sin x)(1 + \cos x) = \frac54$ , and that $(1-\sin x)(1- \cos x)$ can be expressed as $\frac ab - \sqrt c$ for positive integers $a,b,c$ with $a,b$ coprime, find the value of $a+b+c$ .

23 28 21 27

1 solution

Natsir Muhammad
Jun 1, 2015

let (1-sinx)(1-cos) = k

then 5k = sin^2(2x) ........(i)

then 5/4 + k = 2 + sin2x <=> sin2x = k - 3/4 ........(ii)

from (i) and (ii) i got 16k^2 - 104k + 9 = 0

then k = 13/4 - sqrt(10) , yeah :D :D

which mean 13 + 4 + 10 = 27

Plus Minus

1 solution

0 pending reports