Q. If `sintheta` and `log_2a` are rational numbers,then which of the following must be a rational number and why?
Given `sintheta` and `log_2 a` are both rational:
We cannot say that `sin2theta` is rational -- note that `sinpi/6=1/2` which is rational but `sinpi/3=sqrt(3)/2` which is irrational. Thus we can eliminate A.
Since both `sin2theta` and `log_3a` can be irrational, we can eliminate B.
We can eliminate D since `log_3a` is irrational. (`log_3a=(log_2a)/(log_2 3)` and `log_2 3` is irrational.)
Let `sintheta=x/y` with `x,y in ZZ` (x,y both integers) Then `cos2theta=1-2sin^2 theta=1-2(x^2/y^2)` which is certainly rational.
`log_4 a=(log_2 a)/(log_2 4)=(log_2 a)/2` . A rational divided by a rational will be rational.
The answer is C) `cos2theta, log_4 a` will be rational under the conditions stated.