3. trignometrical ratios,functions and identities — MATHS STD 11 Science — Question

$f(\theta ) = r.(\cos \alpha \cos \theta + \sin \alpha \sin \theta ) = 5.\cos (\theta - \alpha )$ 

$\therefore $ The maximum value of $f(\theta ) = 5.1 = 5.$

{Since the maximum value of $\cos (\theta - \alpha ) = 1$}. 

Aliter : As we know that, the maximum value of $a\sin \theta + b\cos \theta $ is $ + \sqrt {{a^2} + {b^2}} $

and the minimum value is $ - \sqrt {{a^2} + {b^2}} $. 

Therefore, the maximum value is $(3\cos \theta + 4\sin \theta ) = + \sqrt {{3^2} + {{( - 4)}^2}} = 5$ 

and the minimum value is $-5.$