i)express the function f(θ)=sinθ + cosθ in the form rsin(θ+α), r>0 and 0≤θ≤≤(π/2) ii)hence find the maximum value of f and the smallest non−negative value of θ at which it occurs.


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Question Number 10746 by okhema last updated on 24/Feb/17

$i) e x p r e s s t h e f u n c t i o n f (θ) = s i n θ + c o s θ i n t h e f o r m r s i n (θ + α), r > 0 a n d 0 ⩽ θ ⩽⩽ \frac{π}{2}$ $i i) h e n c e f i n d t h e m a x i m u m v a l u e o f f a n d$ $t h e s m a l l e s t n o n - n e g a t i v e v a l u e o f θ a t w h i c h i t o c c u r s .$
	Answered by mrW1 last updated on 24/Feb/17

	$i)$ $f (θ) = s i n θ + c o s θ = \sqrt{2} (s i n θ \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} c o s θ) = \sqrt{2 (} s i n θ \cos \frac{π}{4} + c o s θ \sin \frac{π}{4})$ $= \sqrt{2} \sin (θ + \frac{π}{4})$ $i i)$ $m a x . f (θ) = \sqrt{2} w i t h θ = \frac{π}{4}$ $m i n . f (θ) = 1 w i t h θ = 0 o r \frac{π}{2}$
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