If x and y are the maximum and minimum values of sin θ + cos θ, find the value of x2 + y2
If you know that the maximum and minimum value of asinθ + bcosθ are √, and -√,you can directly use this result to arrive at the answer.
We will try to arrive at the answer similar to the derivation of the above result.
Let E = asinθ + bcosθ, If we can express it as k sin (A + B), we can say the minimum and maximum values are -k and +k, where k is a +ve number.
we will expand ksin (A+B) and campare it with asinθ + bsinθ.
ksin(A+B) = asinθ + bcosθ
k[sinAcosB + cosAsinB) = asinθ + bcosθ
If we take A = θ, then we get k cos B = a and Ksin B = b ___________(1)
⇒ cosB = ak and sinB = bk
cos2B + sin2B = a2+b2k2 = 1
⇒ k = √
From (1) we get = cosB = a√
sinB = b√
So we will divide and multiply by √ in asinθ + bcosθ.
⇒ √ (a√sinθ+b√cosθ)
⇒ asinθ + bcosθ = √ (sinθx cosB + cosθsinB)
= √ sin(θ + B)
→ Maximum and minimum values are
√ and - √
In our question a = 1 , b =1
⇒ The minimum and maximum values are −√ and √
⇒ x2 + y2 = 2 + 2 = 4