Block pull, Problems with trig to isolate for theta.

[cambo86]

Homework Statement

What value(s) of θ will result in the block being accelerated 9.0m/s² to the right?

Homework Equations

ƩF_x = ma_x
N = mg - P sin(θ)

The Attempt at a Solution

P cos(θ) - μ_kN = ma_x
P cos(θ) - μ_k(mg - P sin(θ)) = ma_x
P cos(θ) + μ_k P sin(θ) = ma_x + μ_kmg
cos(θ) + μ_k sin(θ) = (1/P)(ma_x + μ_kmg)

From here I can't find a trig identity that will help me isolate for theta.

 

[voko]

$a \cos \theta + b \sin \theta = r (\cos \alpha \cos \theta + \sin \alpha \sin \theta)$, where $r = \sqrt {a^2 + b^2}$ and $\cos \alpha = a/r , \ \sin \alpha = b/r$.

 

[gneill]

Another approach is to replace one of the trig functions (say cos) using the identity sin² + cos² = 1. Rearrange the resulting expression so that squaring both sides eliminates the square root.