Graphs of Sine and Cosine Functions

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Amanda O’Connor’
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1 Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the otation of a line about a point. We detemined that the distance taveled by a point moving all the way aound the cicumfeence of the unit cicle with adius = 1 was equal to 2π, and we defined the adian as the length of the ac on the unit cicle equal to the adius, as shown below. The distance taveled aound the cicumfeence of any cicle is thus equal to 2π adians. On the unit cicle as well as any cicle centeed in a ectangula coodinate system, the tigonometic functions wee defined in tems of the hoizontal and vetical components of a point on the cicle as well as the adius of the cicle. Thus fo the cicle shown below with adius, cicula point P having hoizontal and vetical components a and b, espectively, and x being the distance in adians taveled aound the cicle to point P, the tigonometic functions wee defined as follows:

b sin x = b tan x=, a a 0 sec x=, a a 0 a cos x = a cot x=, b 0 b csc x=, b 0 b We would now like to gaph the tigonometic functions of the ac x as the cicula point moves aound the cicumfeence of the

2 b sin x = b tan x=, a a 0 sec x=, a a 0 a cos x = a cot x=, b 0 b csc x=, b 0 b We would now like to gaph the tigonometic functions of the ac x as the cicula point moves aound the cicumfeence of the cicle. On the gaph, the vetical axis y epesents the value of the tigonometic function and the hoizontal axis epesents the length of the ac x in adians. We may begin by plotting the gaph of the sine function, that is, y = sin x, as shown below.

3 If we stat with x = 0 and move in a counte-clockwise (positive) diection aound the cicle of Fig. 2, the value of x is plotted along the hoizontal axis to the ight of the y-axis in Fig. 3. If we wee to move in a clockwise (negative) diection, the value of x would be plotted to the left of the y-axis. Fo x = 0, the vetical component b of the cicula point P is equal to 0, so b 0 y = sin x = sin 0 = 0 = = Ou fist point on the gaph of y = sin x is thus y = sin 0 = 0. This is the point (0, 0) located at the oigin of the gaph. If the cicula point now moves a quate of the way aound the cicle in a positive diection, ac x equals π/2 adians, the vetical component b of the cicula point is equal to the adius, and π b sin x = sin = = = 1 2 Ou second point on the gaph is thus y = sin π/2 = 1, located at (π/2, 1). Continuing to move anothe quate of the way aound the cicle, x equals π adians, the vetical component b is again equal to 0, and b 0 sin x = sinπ = = = 0 Ou thid point on the gaph is thus y = sin π = 0, located at (π, 0). Continuing anothe quate tun, x equals 3π/2 adians, and the vetical component b is again equal in length to the adius but is now negative. (Remembe fom the pevious sections that the adius is always a positive numbe.) Thus, 3π b sin x = sin = = = 1 2 Ou fouth point on the gaph is then y = sin 3π/2 = 1, located at (3π/2, 1). Continuing aound the final quate of the cicle, x equals 2π adians, and like x = 0, the vetical component b is equal to 0, and

4 b 0 sin x = sin 2π = = = 0 Ou fifth point on the gaph is then y = sin 2π = 0. Having found these five citical points on the gaph of the function y = sin x, we would now like to fill in the gaph between these points, as shown in Fig. 4 below. We should note that the function y = sin x = b/ has a maximum value of 1 at x = π/2, since at that point b is equal to. The magnitude of b can neve exceed, since b is always just the vetical component of the adius. The function has a minimum value of 1 at x = 3π/2, since at that point b is equal to. As the cicula point moves aound the cicle of Fig. 2, the value of the function y = sin x = b/ is dependent only on the value of b, since the value of emains constant. Fo the fist quate of the cicle, the function thus inceases fom 0 to 1 as b inceases fom 0 to. Fo the second quate of the cicle, the function deceases fom 1 to 0 as b deceases fom to 0. Fo the thid quate of the cicle, the function deceases fom 0 to 1 as b deceases fom 0 to. And fo the final fouth quate of the cicle, the function inceases fom 1 to 0 as b inceases fom to 0. Obsevation of the changes in the vetical component b as the cicula point moves aound the cicle of Fig. 2 can explain the shape of the sine function s gaph in Fig. 4. Nea x = 0 and x = π, b (and theefoe y = sin x = b/) is inceasing o deceasing the fastest fo a given change in x, since hee most of the movement of the point is vetical, while nea x = π/2 and x = 3π/2, b is inceasing o deceasing the slowest, since hee most of the movement of the point is hoizontal.

5 Having gone completely aound the cicle one time, we can say we have completed one cycle on the gaph of the function. If we wee to continue aound the cicle a second o moe times, the values of y = sin x would be exactly the same, and the second o thid cycle would look just like the fist cycle. This kind of function is called a peiodic function, since the patten continues to epeat itself. The distance along the hoizontal axis epesenting one cycle is called the peiod of the function. In this case, the peiod of y = sin x is equal to 2π adians. Shown below ae two positive cycles and one negative cycle of y = sin x. Now we can move on to gaph the next tigonometic function, the cosine function y = cos x. This function is simila in all espects to the sine function, except that now we ae focusing on the hoizontal component a of the cicula point P as it moves aound the cicle of Fig. 2 instead of the vetical component b. a Fo x = 0, a is equal to, so y = cos x = cos 0 = 1 = =. π a 0 Fo x = π/2, a is equal to 0, and cos x = cos = = = 0. 2 a Fo x = π, a is equal to, and cos x = cosπ = = = 1. 3π a 0 Fo x = 3π/2, a is equal to 0, and cos x = cos = = = 0. 2 a Finally, fo x = 2π, a is again equal to, and cos x = cos 2π = = = 1.

6 These five points on the gaph of y = cos x ae shown below. Filling in the est of the gaph between these points can be accomplished in the same manne as fo the sine function. The value of y = cos x = a/ as the cicula point moves aound the cicle is simply a function of the hoizontal component a of the cicula point, since the adius emains constant. The shape of the gaph tuns out to be exactly the same as the sine function gaph, the only diffeence in the two being that the cosine gaph stats out at y = 1 fo x = 0 while the sine gaph stated out at y = 0 fo x = 0. The peiod of the two functions is the same, 2π adians, since both functions begin to epeat themselves afte the cicula point finishes one complete evolution. Positive and negative cycles of y = cos x ae shown below. We would now like to expand the basic sine and cosine tigonometic functions to include moe geneal cases of these functions, that is, functions which have been tansfomed in vaious ways.

7 Sine and cosine functions, which ae sometimes said to be sinusoidal in natue, epesent the behavio of many phenomena in the physical wold. These include, but ae cetainly not limited to, sound and adio waves, light ays, othe foms of adiation, altenating electic cuent, and vibations. Instead of the simple basic functions y = sin x and y = cos x, we would like to examine a moe geneal fom of the functions, fo example y = k + A sin (Bx + C). This is still a sine function of x, but the constants allow fo a numbe of tansfomations to the basic function. Remembe fom the pevious section that the maximum value of y = sin x is 1, and this occus at x = π / 2 adians. If we multiply the function by the constant A, whee A is any positive numbe, the maximum value of the function will be A at x = π / 2. If A is a negative numbe, then the function will be eflected about the x-axis. The gaph of y = A sin x can be dawn by multiplying each value of the oiginal function y = sin x by A. The absolute value of the constant A is called the amplitude of the function. Some gaphical examples of the function y = A sin x fo diffeent values of A ae shown below.

8 We next investigate anothe tansfomation of the function by multiplying the value of x by the constant B. In doing this, we ae changing the peiod of the function, not its amplitude. In a nomal cycle, the peiod of the function is equal to 2π adians, since that value of x epesents one complete evolution of a cicula point aound the cicumfeence of a cicle, and beyond that the function begins to epeat itself. If we multiply x by the constant B, the peiod of the function now becomes equal to 2π / B. Fo values of B geate than 1, the peiod is shotened, and fo values of B less than 1, the peiod is lengthened. Some examples of the function y = sin Bx fo diffeent values of B ae shown below. A thid tansfomation of the basic function is obtained by adding the constant C to the value of Bx. When we do this, we shift the entie function to the left o ight, depending on whethe C is positive o negative. If C is positive, the value of Bx + C is geate than Bx, and the function has eached its position fo a given value of x soone than it othewise would have. The patten of the function has thus been shifted to the left to compensate fo the smalle value of x equied to poduce a given y value. The amount of the shift is equal to C if B is equal to 1, but fo othe values B becomes a facto and the shift is equal to C / B. Because a positive value of C poduces a shift to the left (a negative diection), the shift is said to be equal to C / B. The shift is commonly efeed to as a phase shift. Some examples of phase shifting ae shown below.

A final tansfomation of the basic sine function is obtained by adding a constant k to the entie function. This simply tanslates the function vetically up o down by the value of k.

9 A final tansfomation of the basic sine function is obtained by adding a constant k to the entie function. This simply tanslates the function vetically up o down by the value of k. An example of this tanslation is shown below. We have thus discussed fou diffeent tansfomations of the basic sine function. A geneal fom of the sine function which incopoates all of these possible tansfomations is y = k + A sin (Bx + C). Of couse, if k and C ae equal to 0 and A and B ae equal to 1, we ae left with the basic sine function y = sin x and thee ae no tansfomations.

10 We may do exactly the same tansfomations to the basic cosine function y = cos x as we have done to the sine function. As we noted in the pevious section, the patten of the cosine function is exactly the same as the sine function, except thee is a hoizontal shift in the function equal to one-fouth of the peiod. The geneal fom of the cosine function is y = k + A cos (Bx + C).

Section 8.2 Polar Coordinates

Section 8.2 Polar Coordinates Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal