SOLUTION: Find the minimum value of cos<font face="symbol">q</font> + 2sin<font face="symbol">q</font>, if <font face="symbol">q</font> is an element of (-<font face="symbol">p</font>/2, <fo

cosθ + 2sinθ

We will strive to cause this to become a constant times
the right side of the identity 

sin(α+β) = sinα·cosβ + cosα·sinβ

Where β = θ  and 

sin(α+θ) = sinα·cosθ + cosα·sinθ

We wish to find k and α so that 

cosθ + 2sinθ = k·sinα·cosθ + k·cosα·sinθ = k·sin(α+θ)

So we must have 1 = k·sinα  and 2 = k·cosα

or              sinα =   and cosα = 

Since  and 

we construct angle α in a right triangle by choosing α's
opposite side to be 1 and its adjacent side to be 2,
and its hypotenuse to be k.



By the Pythagorean theorem, k² = 2²+1² = 4+1 = 5, so k = 

So

cosθ + 2sinθ = k·sinα·cosθ + k·cosα·sinθ = k·sin(α+θ) 
 
becomes

cosθ + 2sinθ = ·cosθ + ·cosα·sinθ = ·sin(α+θ) 

Since the amplitude of y = ·sin(α+θ) is , its

minimum value is 

Edwin