In the reduction identity :
m sin θ + n cos θ = √(m² + n²) sin(θ + α)
I am having trouble with determining the value of α. Here is an example.
Problem : -7 sin θ - 24 cos θ m = -7 n = -24 Using the above formula : √(-7² + -24²) sin(θ + α) = 25 sin(θ + α)
From here, I use the following identities to attempt to determine α
sin α = n / √(m² + n²)
cos α = m / √(m² + n²)
sin α = -24/25, α = -74° cos α = -7/25, α = 106°
At this point, I have two possible values for α. My textbook states, "α is the smallest possible positive value that satisfies both of these conditions," and lists the value of α as 254°. I'm a bit confused. How did they arrive to that conclusion, and what steps can I take to solve the problem?