1. Modeling with the Sine Function | 2. The Six Trigonometric Functions | 3. Derivatives of Trigonometric Functions | Trigonometric Functions Main Page | "RealWorld" Page | Everything for Calculus | Español |
Answers to Odd-Numbered Exercises |
On the same set of axes, graph the given functions or pairs of functions:
(a) First sketch the curve without any calculator by consulting the discussion in Example 1
(b) Using graphing technology or a calculator to check your sketches.
1. $f(t) = \cos(t); g(t) = -2.5\cos(t)$ | 2. $f(t) = \cos(t); g(t) = 3.1\cos(t)$ |
3. $f(t) = \tan(t); g(t) = -4\tan(t)$ | 4. $f(t) = \cotan(t); g(t) = -2\cotan(t)$ |
5. $f(t) = \sec(t); g(t) = \sec(t - \pi/4)$ | 6. $f(t) = \tan(t); g(t) = \tan(t + \pi)$ |
7. $f(t) = \cosec(t); g(t) = \sin(2t)$ | 8. $f(t) = \cosec(t); g(t) = \sin(-t)$ |
9. $f(t) = 3 + 2\cotan(\pit)$ | 10. $f(t) = 2 - 2\sec(\pit)$ |
11. $f(t) = 4.2\cos(2\pit) + 3$ | 12. $f(t) = 3 - \cos(t-4)$ |
13. $g(x) = 4 - 1.3\cos[2.3(x-4)]$ | 14. $g(x) = 4.5\cos[2\pi(3x-1)] + 7$ |
Model each of the following curves with a sine function.
15. |
16. |
17. |
18. |
19. |
20. |
Some Identities
Starting with the identity $\sin^2x + \cos^2x = 1,$ and then dividing both sides of the equation by a suitable trigonometric function, obtain the following trigonometric identities.The following exercises are based on the addition formulas
$\sin(x + y) = \sin x \cos y + \cos x \sin y$
$\sin(x - y) = \sin x \cos y - \cos x \sin y$
$\cos(x + y) = \cos x \cos y - \sin x \sin y$
$\cos(x - y) = \cos x \cos y + \sin x \sin y$
Applications
31.^{†} Sales Fluctuations† See Exercise 23 in Section 1.
32.^{*} Seasonal Fluctuations* See Exercise 24 in Section 1.
Music Musical sounds exhibit the same kind of periodic behavior as the trigonometric functions. High pitched notes have short periods (typically several thousands per second) while the lowest audible notes have periods of about $1/100$ second. Electronic synthesizers work by superimposing (adding) sinusoidal functions of different frequencies to create different textures. The following exercises, show some examples of superposition can be used to create interesting periodic functions. 33. Saw-Tooth Wave 34. Square Wave
Repeat Exercise 33 using sine functions in place of cosine functions in order to approximate a square wave.
35. Harmony
If we add two sinusoidal functions whose frequencies are exact ratios of each other, the result is a pleasing sound. The following function models two notes an octave apart together with the intermediate fifth.
36. Discord
If we add two sinusoidal functions with similar, but unequal, frequency, the result is a function that "pulsates," or exhibits "beats." (Piano tuners often use this phenomenon to help them tune an instrument.) Graph the function
Communication and Reasoning Exercises
37. Your friend is telling everybody that all six trigonometric exercises can be obtained from the single function $\sin x.$ Is he correct? Explain your answer.
38. Another friend claims that all six trigonometric exercises can be obtained from the single function $\cos x.$ Is she correct? Explain your answer.
39. If weekly sales of a commodity are given by $s(t) = A + B\cos(ωt),$ what is the largest B can be? Explain your answer.
40. Complete the following sentence. If the cost of an item is given by $c(t) = A + B\cos(ω(t-α)),$ then the cost fluctuates by ____ with a period of ____ about a base of ____, peaking at time $t =$ ____.
1. Modeling with the Sine Function | 2. The Six Trigonometric Functions | 3. Derivatives of Trigonometric Functions | Trigonometric Functions Main Page | "RealWorld" Page | Everything for Calculus | Español |
Answers to Odd-Numbered Exercises |
Stefan Waner (matszw@hofstra.edu) | Steven R. Costenoble (matsrc@hofstra.edu) |