Weird functions
Level
pending
Let f(x) = tan x for all x between 0 and 180 degrees and gf(x) = sin 5x. Find the value of a+b+c where g(2) = -(a\sqrt{b})/c where a and c are positive coprime integers and b is not divisible by any perfect square other than 1.
The answer is 168.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Let f(x) = tan x = 2. Hence cos x = 1/sec x = 1/\sqrt{1+ tan^2 x} = 1/\sqrt{1+2^2} = 1/\sqrt{5}. Since sin^2 x= 1- cos^2 x, sin x = 2/\sqrt{5}. Now, sin 5x = sin (3x+2x) = sin 3x cos 2x + sin 2x cos 3x = (3sin x- 4sin^3 x)(1-2sin^2 x) + 2 sin x cos x(4cos^3 x - 3cos x) Substituting the two values of sin x and cos x, we get sin 5x = -38/(25\sqrt{5}) = -(38\sqrt{5})/125. Hence a=b+c = 38+5+125 = 168.