Mac1114 College Trigonometry Project 2 Instructions Either Complete The Project On Separate Or Type Your Answers Usin 1 (41.22 KiB) Viewed 41 times
Mac1114 College Trigonometry Project 2 Instructions Either Complete The Project On Separate Or Type Your Answers Usin 2 (33.86 KiB) Viewed 41 times
MAC1114 College Trigonometry - Project 2 Instructions: Either complete the project on separate or type your answers using MS Word. Label each of your problems clearly and in numerical order. Once your project is complete, save the document as a pdf file and upload your file using the Assignment link in Falcon Online, that is select Assignments and then select Project 1. If you are submitting a handwritten document, you must write NEATLY. If you are submitting a document using MS Word, you must use the Equation Editor correctly. Points will be deducted for work that is not neatly written or the use of incorrect symbols/notation. You need to show all of your work. Part I-Proofs and Formula Derivations Recall the following definitions from algebra regarding even and odd functions: • A function f(x) is even if f(-x) = f(x) for each x in the domain off. • A function f (x) is odd if f(-x) = -f(x) for each x in the domain of f. Also note that the graph of an even function is symmetric about the y-axis and the graph of an odd function is symmetric about the origin. The following proof shows that sine is an odd function. Use it as a model to prove that cosine is an even function and that tangent is an odd function. Statement: Show that sine is an odd function. Proof: Let f(t) = sint and consider the following figure: (x,y) 1 -4
We want to show that f(-t) = -f(t). f(-t) = sin(-t) = -y = -sin t= -f(t) and thus the sine function is odd. Now you should show that cosine is an even function and tangent is an odd function is a similar manner. You should use the above figure in your proof. 1. Statement: Show that cosine is an even function. 2. Statement: Show that tangent is an odd function. 3. Derivation: Derive a formula for cos 38, in terms of cos 8 and sin 8 or just cos 8. Part II - Conceptual Questions 1. True or False? Explain your answer. Because sin(-t) = -sint, it can be said that the sine of a negative angle is a negative number. Part III - Application 1. Answer the following questions. Be sure to show your work. A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by y = sin 2t +cos 2t where y is the distance from equilibrium (in feet) and t is the time (in seconds). a. Use the identity a sinB0 + bcos Be=√√² + b²-sin(Bt+C) where Carctan ().a>0, to write the model in the form y = √a² + b².sin(8t + C). b. State the amplitude of the oscillations of the weight. c. Find the frequency of the oscillations of the weight.
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