How Does the Angle θ Relate to Φ in Static Equilibrium of a Tilted Rod?

[NJJ289]

Homework Statement

The uniform rod in the figure is supported by two strings. The string attached to the wall is horizontal, and the string attached to the ceiling makes an angle of ϕ with respect to the vertical. The rod itself is tilted from the vertical by an angle θ. If ϕ = 29.9°, what is the value of θ?

Homework Equations

T_cc=T_cw

The Attempt at a Solution

I've approached the problem with three forces acting on the object. the two string tensions at the ends of the object, and a gravity force acting on its center of mass.

The center of mass is considered the location of the axis. This let's the gravity force= zero torque and reduces the problem to simply setting the tension-torques equal since the object is in equilibrium.

so I tried

T1rsin(a)=T2rsin(b)

[r is distance to center of mass from end, and a and b are angles between force vectors and position vector r. t1 and t2 are tension forces]

this simplified out to

sin(90+\theta)cos(90-\phi)=sin(90+\phi)... which does not yeild an answer.

I subbed T1=T2cos(90-\theta) to do this since T1 and T2 have cancelling horizontal forces.

I have literally spent hours on this problem and I'm completely stumped. Any assistance is greatly appreciated!

 

[NJJ289]

Yeash! got it... made a small mistake with the angles.

it's cos(90-phi)sin(90+theta)=sin(180-phi+theta) in case anyone was wondering