1985 AIME Problem 9

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of \alpha, \beta, and \alpha + \beta radians, respectively, where \alpha + \beta < \pi. If \cos \alpha, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.