The trigonometric functions are also called circular functions, because they describe relationships between angles on the unit circle. We use different ways to describe trigonometry in order to see how the relationships and equations apply to different situations. When the trigonometric functions are defined as circular, the special values of trigonometric functions can be seen on the unit circle.
If t, the length of the arc from the coordinates 1, 0 equals 0, then sin t equals 0, cos t equals 1, and tangent t equals 0. If a circular function results in division by zero, it is undefined. Therefore, each function has a specific domain. Sines and cosines are defined for all real values of t. Tangents and secants are defined for all real numbers, except when the value of x is 0. All other values of the trigonometric functions can also be seen on the unit circle.
The unit circle is divided into quadrants by the Cartesian coordinates, so the signs of each circular function can be determined by the value of t. If the value of t has a positive value for both x and y, then it lies in Quadrant I. In Quadrant II, only the value of y is positive, and x has a negative value, so sin t and cosecant t will be positive.
All other functions will be negative. Such simple expressions generally do not exist for other angles. Some examples of the algebraic expressions for the sines of special angles are:. Note that while only sine and cosine are defined directly by the unit circle, tangent can be defined as a quotient involving these two:.
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. Every circular function can be derived from the sine and cosine. If you wish, you can research these yourself; they are rarely used today.
What are circular functions? Jan 27, Related questions How do you graph the six trigonometric ratios as functions on the Cartesian plane?
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