The domain of a function is the set of all possible input values. For many familiar functions, the domain is the set of all real numbers. In particular, the domain of any linear or quadratic function is the set of all real numbers. However, for other types of functions we must sometimes exclude certain values from the domain.
The range of a function is the set of all output values for the function. There is an input value that will produce any output we want. There are no restrictions on the input values for this function, so its domain is all real numbers. What about the domain and range of the trigonometric functions? The sine and cosine both include all real numbers in their domains; we can find the sine or cosine of any number.
The range of the tangent is all real numbers. These facts about the three trigonometric functions appear in the Section 6. We can use circular functions of real numbers to describe periodic phenomena. We can use a graph to solve trigonometric equations, or the inverse trig keys on a calculator or computer.
Domain: all real numbers. How do the graphs of the circular functions differ from the graphs of the trigonometric functions of angles in degrees? For Problems 21—26, find an angle in each quadrant, rounded to tenths, with the same reference angle as the angle given in radians. Sketch your solutions on a unit circle.
The unit circle and a set of rules can be used to recall the values of trigonometric functions of special angles. Explain how the properties of sine, cosine, and tangent and their signs in each quadrant give their values for each of the special angles. Unit circle: Special angles and their coordinates are identified on the unit circle. These have relatively simple expressions. 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. Previous Trigonometry of Triangles. Next Periodic and Symmetric Functions. Removing book from your Reading List will also remove any bookmarked pages associated with this title. Are you sure you want to remove bookConfirmation and any corresponding bookmarks?
My Preferences My Reading List. Circular Functions. Adam Bede has been added to your Reading List! For now, we gather these facts into the theorem below. We close this section with a few notes about solving equations which involve the circular functions. Next, it is critical that you know the domains and ranges of the six circular functions so that you know which equations have no solutions. Carl Stitz , Ph. Lorain County Community College.
Generalized Reference Angle Theorem The values of the circular functions of an angle, if they exist, are the same, up to a sign, of the corresponding circular functions of its reference angle. Our next task is to determine in which quadrants the solutions to this equation lie. The latter form of the solution is best understood looking at the geometry of the situation in the diagram below.
Can these lists be combined? Assume that all quantities are defined. Strategies for Verifying Identities Try working on the more complicated side of the identity.
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