Many students feel anxiety when having to deal with trigonometric functions and may want to seek help on homework through online tutoring. These students may have difficulties understanding the six trigonometric functions and their graphs. These functions are sine, cosine, tangent, cosecant, secant, and cotangent. It is also important to understand that these functions do not represent angles themselves but rather functions of angles. In fact, they represent ratios. Online tutoring can be a useful tool not only for help on homework, but also to help clarify the similarities and differences between these functions based on their amplitudes, domains and ranges, periods, horizontal and vertical translations, and vertical asymptotes (when they exist).

It is crucial to remember that these functions are valid only for right-angle triangles. Any help on homework offered by an online tutoring service should focus on the following information concerning trigonometric functions:

The sine of an angle or sin(x) is the length of the side opposite the angle divided by the length of the hypotenuse (or sin(x) = opp./hyp.).

The cosine of an angle or cos(x) is the length of the side adjacent to the angle divided by the length of the hypotenuse (or cos(x) = adj./hyp.).

The tangent of an angle or tan(x) is the length of the side opposite the angle divided by the length of the side adjacent to the angle (or tan(x) = opp./adj.). tan(x) can also be expressed as tan(x) = sin(x)/cos(x).

The cosecant of an angle or csc(x) is the inverse of sin(x). For that reason, it can be represented as the inverse of the sin(x) function as shown above, or csc(x) = hyp./opp.

The secant of an angle or sec(x) is the inverse of cos(x). For that reason, it can be expressed as the inverse of the cos(x) function as shown above, or sec(x) = hyp./adj.

The cotangent of an angle or cot(x) is the inverse of the tan(x) function shown above and can be expressed as cot(x) = adj./opp. Alternatively, it can be represented as cot(x) = cos(x)/sin(x).

Once these functions are defined, the student may need help on homework in solving for the measures of angles or sides of right-angle triangles. It is interesting to note that the graphs of these functions are periodic in nature, meaning that they repeat themselves. Further tutoring can help in understanding that the graphs of y = sin(x) and y = cos(x) are the only ones without vertical asymptotes. Conversely, the graphs of y = tan(x), y = csc(x), y = sec(x), and cot(x) all include vertical asymptotes repeated at regular intervals. One cycle of any of these graphs is referred to as a period. Online tutoring should help clarify that the period is the distance on the x-axis needed for the function to start repeating itself again.

The following information concerning the nature of the six trigonometric functions is often least understood by the student seeking tutoring and should be memorized by him or her in order to facilitate progress. The periods for y = sin(x), y = cos(x), y = csc(x), and y = sec(x) are all 2 pi; the periods for y = tan(x) are y = cot(x) are both pi. Online tutoring can also help the student to distinguish between the graphs of sin(x) and cos(x). The graph of y = sin(x) passes through the origin (0, 0) as an increasing function whereas the graph of y = cos(x) has a maximum at the point (0, 1). Tutoring is essential in helping the student understand the difference between these two graphs especially when more complex functions are covered in class. The graphs of y = tan(x) and y = sec(x) both have repeating vertical asymptotes at (pi/2) + n(pi), where n represents an integer. On the other hand, the graphs of y = csc(x) and y = cot(x) both have repeating vertical asymptotes at intervals of n(pi), where n is an integer.

In all, tutoring and more specifically online tutoring, is an amazing tool that any student can use to learn more about trigonometric functions and their characteristics.