Range of a trignonmetric function I came across this in an Engineering entrance book,
What is the range of this: $a^2 \sin^2 x + b \sin x \cos x + c \cos^2 x$
What is the method to find it? I tried the graph approach but didn't know how to proceed.

$$a^2 \sin^2 x + b \sin x \cos x + c\cos^2 x$$
$$=\frac{a^2 2\sin^2 x + b 2\sin x \cos x + 2c\cos^2 x}2$$
$$=\frac{a^2(1-\cos2x)+b\sin2x+c(1+\cos2x))}2$$
$$=\frac{a^2+c}2+\frac12 \{\cos2x(c-a^2)+b\sin2x\}$$
Now, for $A\cos y+B\sin y=C\sin(y+\theta)$(say) where $C\ge 0$
Expanding and comparing the coefficients, we get $A=C\cos\theta, B=C\sin\theta$ 
squaring and adding we get, $C^2=A^2+B^2,C=\sqrt{A^2+B^2}$
As $-1\le\sin(y+\theta)\le 1, -\sqrt{A^2+B^2}\le C\sin(y+\theta)\le \sqrt{A^2+B^2}$
Here $y=2x,A=b, B=c-a^2$
