When you dial a phone number on your iPhone®, how does the smart phone know which key you have pressed? Dual Tone Multi-Frequency (DTMF), also known as touch-tone dialing, was developed by Bell Labs in the 1960s. The Touch-Tone® system also introduced a standardized keypad layout. After testing 18 different layouts, Bell Labs eventually chose the one familiar to us today, with 1 in the upper-left and 0 at the bottom between the star and the pound keys.
The keypad is laid out in a 4 × 3 matrix, with each row representing a low frequency and each column representing a high frequency.
When you press the number 8, the phone sends a sinusoidal tone that combines a low-frequency tone of 852 hertz and a high-frequency tone of 1336 hertz. The result can be found using sum-to-product trigonometric identities.
IN THIS CHAPTER, we will review basic identities and use those to simplify trigonometric expressions. We will verify trigonometric identities. Specific identities that will be discussed are sum and difference, double-angle and half-angle, and product-to-sum and sum-to-product. Music ...
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