Since the amplitude involves vertical distances, it has no effect on the period of a function, and vice versa. The graph is shown in Figure 5. So in this case, we have. Note: This curve is still sinusoidal despite not being periodic, since the general shape is still that of a "sine wave'', albeit one with variable cycles. So far in our examples we have been able to determine the amplitudes of sinusoidal curves fairly easily.
This will not always be the case. This is sometimes called a combination sinusoidal curve, since it is the sum of two such curves. We can see this in the graph, shown in Figure 5. We can use this as follows:.
In general, a combination of sines and cosines will have a period equal to the lowest common multiple of the periods of the sines and cosines being added. In Example 5. 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.
Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions , we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions. Recall that the sine and cosine functions relate real number values to the x - and y -coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane?
We can create a table of values and use them to sketch a graph. Table 1 lists some of the values for the sine function on a unit circle. Plotting the points from the table and continuing along the x -axis gives the shape of the sine function.
See Figure 2. See Figure 3. Again, we can create a table of values and use them to sketch a graph. Table 2 lists some of the values for the cosine function on a unit circle. As with the sine function, we can plots points to create a graph of the cosine function as in Figure 4.
Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. Figure 5 shows several periods of the sine and cosine functions. Looking again at the sine and cosine functions on a domain centered at the y -axis helps reveal symmetries. As we can see in Figure 6 , the sine function is symmetric about the origin. Now we can clearly see this property from the graph. Figure 7 shows that the cosine function is symmetric about the y -axis.
Again, we determined that the cosine function is an even function. The sine and cosine functions have several distinct characteristics:. As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical.
Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are. Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions.
We can use what we know about transformations to determine the period. Notice in Figure 8 how the period is indirectly related to B. Returning to the general formula for a sinusoidal function, we have analyzed how the variable B B relates to the period. A A represents the vertical stretch factor, and its absolute value A A is the amplitude. The local minima will be the same distance below the midline. Figure 9 compares several sine functions with different amplitudes.
The amplitude is A , A , which is the vertical height from the midline. In addition, notice in the example that. Is the function stretched or compressed vertically? The function is stretched. The negative value of A A results in a reflection across the x -axis of the sine function , as shown in Figure Now that we understand how A A and B B relate to the general form equation for the sine and cosine functions, we will explore the variables C C and D.
Recall the general form:. The value C B C B for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function. The greater the value of C , C , the more the graph is shifted. While C C relates to the horizontal shift, D D indicates the vertical shift from the midline in the general formula for a sinusoidal function. See Figure Any value of D D other than zero shifts the graph up or down.
So the phase shift is. We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before C. If the value of C C is negative, the shift is to the left. Determine the formula for the cosine function in Figure To determine the equation, we need to identify each value in the general form of a sinusoidal function.
We can observe this trend through an example. An understanding of the unit circle and the ability to quickly solve trigonometric functions for certain angles is very useful in the field of mathematics.
Applying rules and shortcuts associated with the unit circle allows you to solve trigonometric functions quickly. The following are some rules to help you quickly solve such problems. The sign of a trigonometric function depends on the quadrant that the angle falls in. Sign rules for trigonometric functions: The trigonometric functions are each listed in the quadrants in which they are positive. Identifying reference angles will help us identify a pattern in these values.
For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. You will then identify and apply the appropriate sign for that trigonometric function in that quadrant. Since tangent functions are derived from sine and cosine, the tangent can be calculated for any of the special angles by first finding the values for sine or cosine.
However, the rules described above tell us that the sine of an angle in the third quadrant is negative. So we have. The functions sine and cosine can be graphed using values from the unit circle, and certain characteristics can be observed in both graphs. So what do they look like on a graph on a coordinate plane?
We can create a table of values and use them to sketch a graph. Again, we can create a table of values and use them to sketch a graph. Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. A periodic function is a function with a repeated set of values at regular intervals. The diagram below shows several periods of the sine and cosine functions. As we can see in the graph of the sine function, it is symmetric about the origin, which indicates that it is an odd function.
This is characteristic of an odd function: two inputs that are opposites have outputs that are also opposites. Odd symmetry of the sine function: The sine function is odd, meaning it is symmetric about the origin.
The graph of the cosine function shows that it is symmetric about the y -axis. This indicates that it is an even function. The shape of the function can be created by finding the values of the tangent at special angles. However, it is not possible to find the tangent functions for these special angles with the unit circle.
We can analyze the graphical behavior of the tangent function by looking at values for some of the special angles. The above points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. At values where the tangent function is undefined, there are discontinuities in its graph. At these values, the graph of the tangent has vertical asymptotes. As with the sine and cosine functions, tangent is a periodic function. This means that its values repeat at regular intervals.
If we look at any larger interval, we will see that the characteristics of the graph repeat. The graph of the tangent function is symmetric around the origin, and thus is an odd function. Calculate values for the trigonometric functions that are the reciprocals of sine, cosine, and tangent.
We have discussed three trigonometric functions: sine, cosine, and tangent. Each of these functions has a reciprocal function, which is defined by the reciprocal of the ratio for the original trigonometric function.
Note that reciprocal functions differ from inverse functions. Inverse functions are a way of working backwards, or determining an angle given a trigonometric ratio; they involve working with the same ratios as the original function.
It can be described as the ratio of the length of the hypotenuse to the length of the adjacent side in a triangle. It is easy to calculate secant with values in the unit circle. Therefore, the secant function for that angle is. It can be described as the ratio of the length of the hypotenuse to the length of the opposite side in a triangle. As with secant, cosecant can be calculated with values in the unit circle.
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