BRCC Verifying Identities & Solving Trigonometric Equations Question

THE Problem Packet Unit 2: Verify Identities, Solve Trigonometric Equations, and Applications All graphs MUST include calculations with supporting explanations (ex: period, horizontal and vertical shifts, asymptotes, domain, range, etc.) and well labeled important axis locations as well as these appearing in the graph during the process of presentation. All simplifying and solving processes used in the process of completing a problem MUST be demonstrated and/or explained, and MUST include all appropriate supporting steps with a presentation demonstration of each process employed. **These problems are tentative and may be subject to any changes by the instructor** 1. A spy hides a secret code signal by transmitting it under a carrier wave by summing the code’s signal function with the carrier signal function. To model this total signal, 𝜋 3𝜋 4 4 use the function 𝑅 (𝑡) = 2 cos2 ( 𝑡) − cos ( 𝑡). a. Verify the validity of the identity: cos(3𝜃 ) = 4 cos3 (𝜃 ) − 3 cos(𝜃 ). b. Use the periods of the two component Cosines in 𝑅(𝑡) to figure out its period. c. Graph 𝑅 (𝑡) using technology and list D&R and key locations for 𝑅 (𝑡). d. Also, with the use of technology, graph 𝑅(𝑡) against the individual signal 𝜋 3𝜋 4 4 functions 𝑦 = 2cos 2 ( 𝑡) and 𝑦 = cos ( 𝑡). Use this graph to try to explain the shape of 𝑅 (𝑡). e. Use an equation to solve for all times when no signal strength is measured from 𝑅 (𝑡). (Hint: You will want to use the identity from part (a) verification) Compare this vs your technology graph to check. 2. A circle centered at point C with a central angle 𝜃 is shown. a. Verify the half angle identities using this circle. 𝜃 1 − cos 𝜃 sin ( ) = ±√ , 2 2 𝜃 1−cos 𝜃 2 1+cos 𝜃 tan ( ) = ±√ = 1−cos 𝜃 sin 𝜃 𝜃 1 + cos 𝜃 cos ( ) = ±√ , 2 2 = sin 𝜃 1+cos 𝜃 b. What values of 𝜃 result in either 𝑥 = 0.6 or 𝑥 = 3.5 ? c. What value of 𝑥 results from 𝑦 = 0.84 ? ̂ and line segment 𝐴𝐵 d. Compute the Area of the region between minor arc 𝐴𝐵 using the situation from part (c). 3. Musical notes with frequency of 𝑓 ℎ𝑧 (hertz=cycles per second) can be modeled as sound waves which are individually given using the formula 𝑁𝑓 (𝑡) = 𝐴 sin(2𝜋𝑓𝑡). When more than one note is played simultaneously then the sound is modeled by summing the individual frequency functions. 𝐴−𝐵 a. Prove the sum to product identity: sin(𝐴) + sin(𝐵) = 2 cos ( 𝐴+𝐵 ) sin ( 2 ). 2 b. Use that identity to simplify the model of the sum of two notes, 𝐺2 and 𝐴2 based on a tuning scale for 𝐴4 = 440ℎ𝑧, when played together 𝑀(𝑡) = 𝑁98 (𝑡) + 𝑁110 (𝑡). (Just use volume =1 for both frequencies) c. Using technology graph the result from part (b) on the same grid as a graph of 𝐴−𝐵 just the cosine portion, 𝐵(𝑡) = 2 cos ( ), of your simplification. Notice the 2 “beat” or “packet” pattern demonstrated, comment on how that relates to the period given in this cosine portion of the function. What does the sine portion represent? d. Make note on your graph of how many times per “beat” or “packet” the two graphs of the sound mix, 𝑀(𝑡), and the Beat wave, 𝐵(𝑡), intersect. Use an equation to solve for all the moments in time where these two intersect. Do you think you hear all these tiny peaks, just the packet, or both?

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