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