Graphing in the Entire Coordinate Plane
Graphing in the Entire Coordinate Plane
The functions sine, cosine, and tangent are commonly graphed in the coordinate plane, with x representing the measure of an angle (the x units are usually given in radians) and y measuring the value of a given trigonometric function at that angle. The best way to see this is to study the graphs themselves. In the image below, (a) is the graph of y = sin x; (b) is y = cos x; and (c) is y = tan x.
These graphs make evident a number of important characteristics of trigonometric functions.
The domain of a function is simply the x values for which the function can be calculated. In the case of the trigonometric functions, the input is an angle measure, and the output is a ratio (like opposite/ hypotenuse, for example).
The domain of a trigonometric function can be seen in its graph: it is the set of all x values for which the function is defined. For sine and cosine, the domain is the set of real numbers, because any angle measure has a sine and a cosine; there are no values of x for which sine or cosine doesn’t produce a y value.
The graph of the tangent function, however, tells a different story. It shows certain x values, for which the tangent is undefined. These undefined points occur when the cosine is zero, since tan x = sin x/cos x, and division by zero is undefined. The x-values for which tangent is undefined show up on its graph as vertical dotted lines every 180º, such that x = n(180º) + 90º, where n is an integer. For example, the tangent function is undefined at the x value 2(180º) + 90º = 450º.
Like the domain, the range of the trigonometric functions can be seen from their graphs. The range of a function is the set of all possible values of the function. In other words, the range is the set of all y values of the function.
The range of sine and cosine, as you can see in its graph or by analyzing the unit circle, is –1 ≤ y ≤ 1. The graphs of these two functions never rise above 1 or fall below –1 and every point on the unit circle has a x and y value between –1 and 1. Occasionally, you may see a question in which the answer choices are possible values of sine or cosine. If any of them are greater than 1 or less than –1, you can eliminate them.
The range of tangent is the set of real numbers. To see why there are no bounds on the value of tangent, recall that the denominator (cos ) can get arbitrarily close to zero, making the quotient get infinitely large.
The chart below summarizes what has been discussed in the previous few paragraphs. We have also included the ranges and domains of the other three trigonometric functions.
Periodic Functions
Sine, cosine, and tangent are all periodic functions, meaning that their values repeat on a regular interval. This regular interval is called the function’s period. Speaking more technically, the period of a function is the smallest domain containing a full cycle of the function. Take a look at the periods for sine, cosine, and tangent:
  • For y = sin x and y = cos x, the period is 2π radians. This means that every 360º, the values of sine and cosine repeat themselves. For example, trigonometric functions of 0 and 2π radians produce the same values.
  • For y = tan x, the period is π radians. Thus, the tangents of 0º and 180º are equal.
If a trigonometric function contains a coefficient in front of x, its period changes. In general, the period of y = f(bx) is the normal period of f divided by b. For example, the period of y = sin 1/4 x = 2 radians1 /4 = 8π radians.
Another useful property of the sine and cosine curves (not tangent, though) is amplitude. The figure below shows the amplitude, a, for the sine and cosine functions:
The amplitude of the sine and cosine functions is half the vertical distance between its minimum value and its maximum value. The amplitude of y = sin x and y = cos x is 1 because the minimum and maximum values of these functions are –1 and 1, respectively, and half the vertical distance between these values is 1. The tangent graph has no amplitude, because the tangent function has no minimum or maximum value. In general, the amplitude of the trigonometric function y = af(x) is |a|. The amplitude of 1 /3 cos x is 1/3.
Here is an example of the type of problem you might see concerning the graphs of the trigonometric functions.
What is the period and amplitude of the function y = 4sin 3x?
As we just discussed, the period of y = f(bx) is the normal period of f divided by b. For the sine function, the normal period is 360o. In this example, b = 3, so the period of this function is 3603 = 120o. In general, the amplitude of the sine function y = af(x) is |a|. In this particular case, a = 4. So, the amplitude is 4.
Here’s another example problem:
What is the period and amplitude of the function graphed below?
To answer this question, you simply have to read the distances off the graph. The function, which appears to be either a sine graph or a cosine graph, repeats itself every 180o. Its period is therefore 180o. Its minimum and maximum values are ±1 /2, so its amplitude is 1/ 2.
To handle any question about the graphs of trigonometric functions, you should be able to answer questions about period and amplitude based on the equation or graph of a given function.
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