Expert: Chen Min - 5/7/2008

Question
if sinx=3cosx, what is the value of sinxcosx.

Where do I start?

Answer
A very simple way is to get tan value of x;

tanx = sinx / cosx = 3

So, sinxcosx = 0.5 sin2x = 0.5* 2tanx / [1 + (tanx)^2]
            = 0.3

This is done by the first of the following identities:

sin2α=2tanα/[1+(tanα)^2]

cos2α=[1-(tanα)^2]/[1+(tanα)^2]

tan2α=2tanα/[1-(tanα)^2]

They are useful because they associate sin, cos, and tan values to tan alone, so as to manage and process them better.

An easier alternative is this:

As sinx = 3cosx
  sinx*cosx = 3(cosx)^2                   (1)
            = 3[1 - (sinx)^2]
            = 3[1 - 9(cosx)^2]
            = 3 - 27(cosx)^2              (2)
Hence from (1) and (2),
    30(cosx)^2 = 3
    sinxcosx = 30(cosx)^2 / 10 = 0.3
The key is to constantly substitute the unknown with the known.

Hope this helps:)