Maximum value of $2(\sin A + \sin B) - 2\cos C$

Question

What is the maximum value of $2(\sin A + \sin B) - 2\cos C$ in a triangle $ABC$? Can anyone help?

Answer 1

Let $y=2(\sin A+\sin B)-2\cos C=4\cos\dfrac C2\cos\dfrac{A-B}2-2\left(2\cos^2\dfrac C2-1\right)$

$$\iff4\cos^2\dfrac C2-4\cos\dfrac C2\cos\dfrac{A-B}2+y-2=0$$

Now as $\cos\dfrac C2$ is real, the discriminant must be $\ge0$

i.e., $$\left(4\cos\dfrac{A-B}2\right)^2\ge16(y-2)$$

$$\iff y-2\le\cos^2\dfrac{A-B}2\le1$$ which occurs if $A=B$ as $0<A,B<\pi$

consequently $\cos\dfrac C2=\dfrac12\iff C=\dfrac{2\pi}3$ as $0<C<\pi$