Solve -mu_{A}m_{A}gcosθ-m_{A}g[frac{m_{A}(sinθ-mu_{A}cosθ)+m_{A}(sinθ-mu_{B}cosθ)}{(m_{A}+m_{A})}]

$$- \mu _ { A } m _ { A } g \cos \theta - m _ { A } g [ \frac { m _ { A } ( \sin \theta - \mu _ { A } \cos \theta ) + m _ { A } ( \sin \theta - \mu _ { B } \cos \theta ) } { ( m _ { A } + m _ { A } ) } ]$$

Factor

$$\frac{gm_{A}\left(\mu _{B}\cos(\theta )-\mu _{A}\cos(\theta )-2\sin(\theta )\right)}{2}$$

Solution Steps

Combine m_{A} and m_{A} to get 2m_{A}.

$$factor(\left(-\mu _{A}\right)m_{A}g\cos(\theta )-m_{A}g\times \frac{m_{A}\left(\sin(\theta )-\mu _{A}\cos(\theta )\right)+m_{A}\left(\sin(\theta )-\mu _{B}\cos(\theta )\right)}{2m_{A}})$$

Factor the expressions that are not already factored in \frac{m_{A}\left(\sin(\theta )-\mu _{A}\cos(\theta )\right)+m_{A}\left(\sin(\theta )-\mu _{B}\cos(\theta )\right)}{2m_{A}}.

$$factor(\left(-\mu _{A}\right)m_{A}g\cos(\theta )-m_{A}g\times \frac{m_{A}\left(-\mu _{A}\cos(\theta )-\mu _{B}\cos(\theta )+\sin(\theta )+\sin(\theta )\right)}{2m_{A}})$$

Cancel out m_{A} in both numerator and denominator.

$$factor(\left(-\mu _{A}\right)m_{A}g\cos(\theta )-m_{A}g\times \frac{-\mu _{A}\cos(\theta )-\mu _{B}\cos(\theta )+\sin(\theta )+\sin(\theta )}{2})$$

Combine \sin(\theta ) and \sin(\theta ) to get 2\sin(\theta ).

$$factor(\left(-\mu _{A}\right)m_{A}g\cos(\theta )-m_{A}g\times \frac{-\mu _{A}\cos(\theta )-\mu _{B}\cos(\theta )+2\sin(\theta )}{2})$$

Express m_{A}\times \frac{-\mu _{A}\cos(\theta )-\mu _{B}\cos(\theta )+2\sin(\theta )}{2} as a single fraction.

$$factor(\left(-\mu _{A}\right)m_{A}g\cos(\theta )-\frac{m_{A}\left(-\mu _{A}\cos(\theta )-\mu _{B}\cos(\theta )+2\sin(\theta )\right)}{2}g)$$

Use the distributive property to multiply m_{A} by -\mu _{A}\cos(\theta )-\mu _{B}\cos(\theta )+2\sin(\theta ).

$$factor(\left(-\mu _{A}\right)m_{A}g\cos(\theta )-\frac{-m_{A}\mu _{A}\cos(\theta )-m_{A}\mu _{B}\cos(\theta )+2m_{A}\sin(\theta )}{2}g)$$

Express \frac{-m_{A}\mu _{A}\cos(\theta )-m_{A}\mu _{B}\cos(\theta )+2m_{A}\sin(\theta )}{2}g as a single fraction.

$$factor(\left(-\mu _{A}\right)m_{A}g\cos(\theta )-\frac{\left(-m_{A}\mu _{A}\cos(\theta )-m_{A}\mu _{B}\cos(\theta )+2m_{A}\sin(\theta )\right)g}{2})$$

Use the distributive property to multiply -m_{A}\mu _{A}\cos(\theta )-m_{A}\mu _{B}\cos(\theta )+2m_{A}\sin(\theta ) by g.

$$factor(\left(-\mu _{A}\right)m_{A}g\cos(\theta )-\frac{-m_{A}\mu _{A}\cos(\theta )g-m_{A}\mu _{B}\cos(\theta )g+2m_{A}\sin(\theta )g}{2})$$

Factor out \frac{1}{2}.

$$\frac{m_{A}g\left(-2\mu _{A}\cos(\theta )-\left(-\mu _{A}\cos(\theta )-\mu _{B}\cos(\theta )+2\sin(\theta )\right)\right)}{2}$$

Evaluate

$$\frac{gm_{A}\left(\mu _{B}\cos(\theta )-\mu _{A}\cos(\theta )-2\sin(\theta )\right)}{2}$$