## ExercisesforSection 2: The Six Trigonometric Functions

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. $f(t) = cotan(t); g(t) = -2cotan(t)$ 5. $f(t) = \sec(t); g(t) = \sec(t - π/4)$ 6. $f(t) = \tan(t); g(t) = \tan(t + π)$ 7. $f(t) = \cosec(t); g(t) = \sin(2t)$ 8. $f(t) = \cosec(t); g(t) = \sin(-t)$ 9. $f(t) = 3 + 2cotan(πt)$ 10. $f(t) = 2 - 2\sec(πt)$

Use the conversion formula $\cos x = \sin(π/2 - x)$ to replace each of the following by a sine function:

 11. $f(t) = 4.2\cos(2πt) + 3$ 12. $f(t) = 3 - \cos(t-4)$ 13. $g(x) = 4 - 1.3\cos[2.3(x-4)]$ 14. $g(x) = 4.5\cos[2π(3x-1)] + 7$

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.
21. $\sec^2x = 1 + \tan^2x$   22. $\cosec^2x = 1 + cotan^2x$

The following exercises are based on the addition formulas

$\sin(x + y) = \sin x \cos y + \cos x \sin y$
$\sin(x - y) = \sin x \cos y - \cos x \sin y$
$\cos(x + y) = \cos x \cos y - \sin x \sin y$
$\cos(x - y) = \cos x \cos y + \sin x \sin y$

23. Calculate $\sin(π/3),$ given that $\sin(π/6) = 1/2$ and $\cos(π/6) = (√3)/2$

24. Calculate $\cos(π/3),$ given that $\sin(π/6) = 1/2$ and $\cos(π/6) = (√3)/2$

25. Use the formula for $\sin(x+y)$ to obtain the identity $\sin(t + π/2) = \cos t$

26. Use the formula for $\cos(x+y)$ to obtain the identity $\cos(t - π/2) = \sin t$

27. Show that $\sin(π - x) = \sin x.$

28. Show that $\cos(π - x) = -\cos x.$

29. Use the addition formulas to express $\tan(x+π)$ in terms of $\tan(x).$

30. Use the addition formulas to express $cotan(x+π)$ in terms of $cotan(x).$

Applications

31. Sales Fluctuations
Sales of General Motors cars and light trucks in 1996 fluctuated from a high of $95 billion in October (t = 0) to a low of$80 billion in April (t = 6) Construct a cosine model for the monthly sales s(t) of General Motors.

† See Exercise 23 in Section 1.

32.* Seasonal Fluctuations
Sales of Ocean King Boogie Boards fluctuate sinusoidally from a low of $50$ units per week each February 1 $(t = 1)$ to a high of $350$ units per week each August 1 $(t = 7).$ Use a cosine function to model the weekly sales $s(t)$ of Ocean King Boogie Boards, where t is time in months.

* 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
(a) Graph the following functions in a window with $-7 ≤ x ≤ 7$ and $-1.5 ≤ y ≤ 1.5.$
$y_1 = $$\frac{2}{π} \cos x y_3 =$$\frac{2}{π}$$\cos x +$$\frac{2}{3π}$$\cos 3x y_5 =$$\frac{2}{π}$$\cos x +$$\frac{2}{3π}$$\cos 3x +$$\frac{2}{5π}$$\cos 5x$

(b) Now give a formula for $y_{11}$ and graph it in the same window.

(c) How would you modify $y_{11}$ to approximate a saw-tooth wave with an amplitude of $3$ and a period of $4π$?

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.

$y = \cos(x) + \cos(1.5x) + \cos(2x).$

Graph this function in the window $0 ≤ x ≤ 20$ and $-3 ≤ y ≤ 3,$ and estimate the period of the resulting wave.

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

$y = \cos(x) + \cos(.9x)$

in the window $-50 ≤ x ≤ 50$ and $-2 ≤ y ≤ 2,$ and estimate the period of the resulting wave.

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 $t =$ ____.

We would welcome comments and suggestions for improving this resource. Mail us at:
 Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: September, 1996