When two sinusoidal waves travel through the same medium, a third wave is formed that is the sum of the two original waves. If the two waves have slightly different frequencies, the sum of the waves results in an interference pattern known as a beat. Musicians use this idea when tuning an instrument with the aid of a tuning fork. If the instrument and the tuning fork play the same frequency, no beat is heard. Suppose two waves given by the functions, y₁ = 3 cos(ω₁t ) and y₂ = 3 cos(ω₂t ) where ω₁ > ω₂ pass through the same medium, and each has a maximum at t = 0 sec.

(a) How long does it take the sum function y₃ = y₁ + y₂ to equal 0 for the first time?

(b) If the periods of the two functions y₁ and y₂ are T₁ = 19 sec and T₂ = 20 sec, respectively, find the first time the sum y₃ = y₁ + y₂ = 0.

(c) Use the values from part (b) to graph y3 over the interval 0 ≤ x ≤ 600. Do the waves appear to be in tune?