Explain why Sin of theta= -The square root of 1-cos^2 theta is not an identity, using either graphical or numerical reasoning.

Answer 1

If hypotenuse is 1
opposite = sin T
adjacent = cos T

1^2 = (sin T) ^2 + (cos T)^2 as we know
and
sin^2 T = 1 - cos^2 T
sin T = +/- sqrt (1-cos^2 T)
well, that is not much of an identity if the sign is arbitrary.
BUT
if we use the negative root then
we know that cos^2 T is always <1 so
sin T could be negative even if T were + 30 degrees