On a TouchTone phone each button produces a unique sound The

On a Touch-Tone phone, each button produces a unique sound. The sound produced is the sum of two tones, given by tin (2xLt) and tin (2xHt), where L and H are the low and high frequencies (cycles per second) shown on the illustration. For example. If you touch 0. The low frequency is 941 cycles per second and the high frequency is 1336 cycles per second. The sound emitted by touching 0 is y = sin [2x(941)t] + sin (2x( 1336)t]. (a) Write this sound as a product of sines and/or cosines. (b) Graph the sound emitted by touching 0. (c) Use a graphing calculator to determine an upper bound on the value of y. Enter your answer in the answer box and then click Check Answer. (a) write this sound as a product of sines and/or cosines. y =

Solution

y = sin [ 2pi (941)t ] + sin [ 2pi (1336)t]

applying sum to product formula

sin a + sin b = 2 sin (a+b)/2 cos (a-b)/2

therefore,

sin [ 2pi (941)t ] + sin [ 2pi (1336)t] = 2 sin (2pi (941)t + 2pi (1336)t )   cos (2pi(941)t - 2pi(1336)t) /2

2 (sin (2277 pi*t ) cos (-395 pi*t )