The cosine and sine functions satisfy the following properties of symmetry:Ĭos ( − θ ) = cos ( θ ) sin ( − θ ) = − sin ( θ ). Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the The graph is not horizontally stretched or compressed, so B = 1 B = 1 and the graph is not shifted horizontally, so C = 0.
Also, the graph is reflected about the x-axis so that A = − 0.5. Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so | A | = 1 2 = 0.5. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. The greatest distance above and below the midline is the amplitude. This value, which is the midline, is D D in the equation, so D = 0.5. We can see that the graph rises and falls an equal distance above and below y = 0.5. Since the cosine function has an extreme point for x = 0, x = 0, let us write our equation in terms of a cosine function. When x = 0, x = 0, the graph has an extreme point, ( 0, 0 ). The graph could represent either a sine or a cosine function that is shifted and/or reflected. Y = A sin ( B x − C ) + D y = A cos ( B x − C ) + D y = A sin ( B x − C ) + D y = A cos ( B x − C ) + D Figure 9 compares several sine functions with different amplitudes. If | A | < 1, | A | < 1, the function is compressed. For example, the amplitude of f ( x ) = 4 sin x f ( x ) = 4 sin x is twice the amplitude of f ( x ) = 2 sin x. If | A | > 1, | A | > 1, the function is stretched. The local minima will be the same distance below the midline. The local maxima will be a distance | A | | A | above the horizontal midline of the graph, which is the line y = D y = D because D = 0 D = 0 in this case, the midline is the x-axis. A A represents the vertical stretch factor, and its absolute value | A | | A | is the amplitude. Now let’s turn to the variable A A so we can analyze how it is related to the amplitude, or greatest distance from rest. Returning to the general formula for a sinusoidal function, we have analyzed how the variable B B relates to the period. ĭetermine the period of the function g ( x ) = cos ( x 3 ). Notice in Figure 8 how the period is indirectly related to | B |. If f ( x ) = sin ( x 2 ), f ( x ) = sin ( x 2 ), then B = 1 2, B = 1 2, so the period is 4 π 4 π and the graph is stretched.
If f ( x ) = sin ( 2 x ), f ( x ) = sin ( 2 x ), then B = 2, B = 2, so the period is π π and the graph is compressed. For example, f ( x ) = sin ( x ), f ( x ) = sin ( x ), B = 1, B = 1, so the period is 2 π, 2 π, which we knew. If | B | > 1, | B | > 1, then the period is less than 2 π 2 π and the function undergoes a horizontal compression, whereas if | B | < 1, | B | < 1, then the period is greater than 2 π 2 π and the function undergoes a horizontal stretch. In the general formula, B B is related to the period by P = 2 π | B |. We can use what we know about transformations to determine the period.
Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. Y = A sin ( B x − C ) + D and y = A cos ( B x − C ) + D y = A sin ( B x − C ) + D and y = A cos ( B x − C ) + D Determining the Period of Sinusoidal Functions Table 1 lists some of the values for the sine function on a unit circle. We can create a table of values and use them to sketch a graph. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. In this section, we will interpret and create graphs of sine and cosine functions. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. Light waves can be represented graphically by the sine function. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow. Instead, it is a composition of all the colors of the rainbow in the form of waves. White light, such as the light from the sun, is not actually white at all. Figure 1 Light can be separated into colors because of its wavelike properties.