Problem:
If sinx=3cosx then what is sinxcosx?
Answer Choices:
A. 61
B. 51
C. 92
D. 41
E. 103
Solution:
If sinx=3cosx then tanx=3. From the figure we conclude that
sinxcosx=103⋅101=103
for any acute angle x. If x′ is another angle with tanx′=3,x′−x is a multiple of π. Thus
sinx′=±103,cosx′=±101
So sinx′cosx′ is still 3/10 (since sinx′ and cosx′ have the same sign ).
OR
Multiplying the given equation first by sinx and then by cosx yields
sin2x=3sinxcosxcos2x=(1/3)sinxcosx
Adding gives
1=(10/3)sinxcosx
so sinxcosx=3/10.
