Projectile Motion with Relative Motion (w/o Solver)

[bartersnarter]

Homework Statement

Homework Equations

v_x = 70*cos(α) + 8.333 (30 km/h = 8.333 m/s)
v_y = 70*sin(α)
v_x*t = 350
v_y*t = 60
-(1/2)*g*t^2 + v_y*t + 2.5 = 60

The Attempt at a Solution

This problem is conceptually very simple for me, but I can't solve it without using a solver like Wolfram Alpha. First, I simply rearranged v_x*t = 350 into t = 350/v_x = 350/(70*cos(α) + 8.333). Then, I plugged this t into the -(1/2)*g*t^2 + v_y*t + 2.5 = 60 equation and I got an equation with only the angle α as the unknown. After moving the equation around so that there are no denominators, I get:
600825.5 + 204167*sin(α) + 1715000*sin(α)*cos(α) = 281750*cos²(α) + 67083*cos(α) + 3993

Is there a way to solve this equation without using a solver? Is there another method entirely which I'm missing? I have the right answers, which are α = 31.32° and 74.28°.

 

[haruspex]

Homework Statement

Homework Equations

v_x = 70*cos(α) + 8.333 (30 km/h = 8.333 m/s)
v_y = 70*sin(α)
v_x*t = 350
v_y*t = 60
-(1/2)*g*t^2 + v_y*t + 2.5 = 60

The Attempt at a Solution

This problem is conceptually very simple for me, but I can't solve it without using a solver like Wolfram Alpha. First, I simply rearranged v_x*t = 350 into t = 350/v_x = 350/(70*cos(α) + 8.333). Then, I plugged this t into the -(1/2)*g*t^2 + v_y*t + 2.5 = 60 equation and I got an equation with only the angle α as the unknown. After moving the equation around so that there are no denominators, I get:
600825.5 + 204167*sin(α) + 1715000*sin(α)*cos(α) = 281750*cos²(α) + 67083*cos(α) + 3993

Is there a way to solve this equation without using a solver? Is there another method entirely which I'm missing? I have the right answers, which are α = 31.32° and 74.28°.

You could turn it into a quartic in cos(α), but you would not be that much better off.