Express −3sin x − 4 cos x in the form k sin (x + φ ). Round φ to the nearest degree.
k = √((-3)^2 + (-4)^2)) = 5
sin φ = -3/5
cos φ = -4/5
φ = sin^(-1)(-3/5) ≈ -37 degrees
Final answer: 5sin(x - 37°)
Hint: To check your answer use the the formula efor addition of angles to check your answer. Do you know how to calculate cos(arcsin(x))? If not construct a right angled triangle to get the value as a function of x.
Hi Chiro, I wasn't sure how to check, but I tried using the addition of angles, looking over my teacher's notes, I tried following along his lines, but I don't understand. This is my guess work:
−3sin x − 4 cos x
5 (-3sinx - 4 cosx)
= 5 (cosφsinx - sinφcosx)
= 5 sin (x - φ)
= 5 sin (x - 37°)
I would lust like to approach the question like this:
−3sin x − 4 cos x = -5 [ 3/5 sin x + 4/5 cos x ] = -5 [ sinx cos (phi) + cos x sin ( phi) ] = -5 sin ( x + phi) where cos (phi ) = 3/5 and sin ( phi ) = 4/5 Now it is in the required for where k = -5 and phi = arcsin (4/5)