MATH117 Lecture Notes - Lecture 9: Sine Wave, Angular Frequency, Inverse Trigonometric Functions

• School: University of Waterloo
• Department: Mathematics
• Course: MATH117
• Professor: David Harmsworth

Document Summary

The pythagorean identity and the double-angle formulas are of critical importance in calculus, for reasons we"ll explore later (they are essential tools for the evaluation of some common types of integrals). The sum-of-angle formulas are also extremely important, but for a di erent reason, which we will discuss now. This is, in fact, still a sine wave. It has the same angular frequency as the input, but a di erent amplitude and phase. That is, it can be re-written in the form1 g(t) = a sin( t + ). The key is the double angle formula, sin( 1+ 2) = sin 1 cos 2+ sin 2 cos 1. A sin( t + ) = a sin t cos + a cos t sin , and we can work backwards from this to determine what the values of a and must be. Example: express y = 5 cos 2t 3 sin 2t in the form y = a sin(2t + ).