A Guide to Evaluating Trigonometric Functions for Common Angle Values

Transcription

1 A Guide to Evaluating Trigonometric Functions for Common Angle Values A student is often faced with the task of evaluating one of the six common trigonometric functions for some common multiple of pi, usually integer multiples of,,,, or. A student who has completed precalculus with trigonometry should be able to produce an exact evaluation of the trigonometric function for these angles without resorting to the use of a calculator. This document provides a simple methodology for producing such evaluations. The table on the next page represents our goal the student should be able to readily reconstruct this somewhat intimidating table. A blank table is provided on the following page so the student has a template for practice. The remainder of this document walks the student through the elementary steps that allow him to reconstruct the values of the six trigonometric for the common angle values of the first four quadrants. TrigTable 5 September

2 VALUES OF THE SIX TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES sinθ cosθ tanθ Udf Udf cotθ Udf Udf Udf secθ Udf Udf cscθ Udf Udf Udf Udf means Undefined. TrigTable 5 September

3 VALUES OF THE SIX TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES sinθ cosθ tanθ cotθ secθ cscθ Use copies of this page to practice learning the values of the trigonometric functions. TrigTable 5 September

4 How to Learn the Table of Trigonometric Values The table has rows and 8 columns (the θ = column is a repeat of the θ = column) for a total of table values. It may seem to be a superhuman effort to memorize such a table. Indeed, it is much better to learn the patterns present in the table and use these patterns to reconstruct individual entries. The remainder of this document will help you to learn the table patterns. We first notice that the tangent, cotangent, secant, and cosecant functions are derived from the sine and cosine functions. Therefore, if we learn the first two table rows, we will be able to reconstruct the remaining four rows. We have cut our work by nearly /! (I use the word nearly because there is some arithmetic involved in calculating the remaining values.) We next notice that the cosine function takes the same values as the sine function, but the values are shifted with respect to the angle θ. If we learn that pattern, it suffices to learn just the first row of the table. Next, we observe that the sine function repeats its values from quadrant to quadrant, with the occasional change of sign. Since all four quadrants are represented, it suffices to remember the values of the sine function for only the first quadrant. Therefore, if we can remember the sine function for 5 values of θ along with some rules for populating the remainder of the table, we have all table values! There is one more bit of work I neglected to mention it is necessary to remember the common values of the angle parameter θ. However, patterns once again come to our rescue; it is necessary to learn only approximately different numbers. Radian Values of Common First Quadrant Angles The first quadrant angles of interest have values of,,,, and ; call these the common first quadrant angles. These radian measures correspond to degree values of,, 5,, and 9 degrees, respectively. While it is permissible to interpret radian measures in terms of the corresponding degree values, the student should quickly learn to think in terms of radian measure. Note that each first quadrant common angle fraction has as a numerator. The denominators of the sequence of fractions are decreasing exactly what is required for the values of the fractions to form an increasing sequence. The denominators are simple integers and must be learned. Note also that the first quadrant common angle values are symmetric about in that the values can be paired in such a way so that the sum of each pair is. That is, + = and + =. What about? It can be paired with itself: + =. TrigTable 5 September

5 Values of the Sine in the First Quadrant The following table shows a simple pattern for remembering the values of the sine function for the angle values described in the prior section. sinθ sinθ Note that the values in the second row for the sine function have the same value as the corresponding value in the first row. Therefore, if one can begin at and count whole numbers to the value, one has everything required to reproduce values of the sine function for the common first quadrant angle values. TrigTable 5 5 September

6 Values of the Cosine for the Common Angles in the First Quadrant We may use the identity cosθ = sin θ and the symmetry of the first quadrant common angles about to develop the table to include values of the cosine function for the first quadrant common angles. Using the symmetry of the first quadrant common angles about, we see that the cosine values repeat the sine values, but in decreasing order. sinθ cosθ TrigTable 5 September

7 Values of the Tangent for the Common Angles in the First Quadrant The tangent function is defined as the ratio of the sine and cosine functions. This makes extending the table to include the values for the tangent function in the first quadrant relatively simple: sinθ cosθ tanθ Udf It may be instructive to review the arithmetic required for rationalization of the denominator. The arithmetic for tan is developed: tan = = = = = Note that the three tangent values for,, and form a geometric sequence with tan = tan and tan = tan permitted (Udf means undefined.) as the common ratio; that is,. Note also that is not in the domain of the tangent function as division by is not TrigTable 7 5 September

8 Extending the Table to All Four Quadrants The first step in extending the table to quadrants II, III, and IV is determining the values of the common angles for those quadrants. As it happens, these values can be easily derived from the corresponding first quadrant common angle values. The portion of the table that lists the angle values appears below: Quadrant Boundaries The quadrant boundaries appear at,,,, and. Note that the sequence of denominators,,, and, repeats within each quadrant, but the pattern reverses descending to ascending to descending - at each quadrant boundary. Thus, the student should be able to partially reconstruct the first line of the table as follows: Second Quadrant We previously noted the coefficient of pi in the numerator was for the first quadrant common angles. There are similar patterns for each of the three remaining quadrants. The coefficient of pi in the numerator of the second quadrant common angles is always one less than the value of the denominator. That is, for we have =. For angle values. we have =, etc. It is a simple matter to complete the second quadrant common 5 TrigTable 8 5 September

9 Third Quadrant The coefficient of pi in the third quadrant is always one more than the value of the denominator. That is, for 7 is a simple matter to complete the third quadrant common angle values. we have 7 = +. It Fourth Quadrant Finally, the coefficient of pi in the fourth quadrant is one less than twice the value of the denominator. That is, for 7 7 =. This allows us to easily complete the sequence of common angle values. we have TrigTable 9 5 September

10 Values of the Sine, Cosine, and Tangent Functions for All Four Quadrants The values of the sine, cosine and tangent functions are readily extended to the remaining three quadrants by keeping track of the sign of each function in the respective quadrants. There is a simple mnemonic device for remembering which of the three functions is positive in each of the four quadrants: ASTC (or All Students Take Calculus). Each of the four letters represents one quadrant, A for I, S for II, T for III, and C for IV. The A mean All all three functions are positive in the first quadrant. S represents the sine function only the sine function is positive in Quadrant II. T represents the tangent function only the tangent function is positive in Quadrant III. Finally, C represents the cosine function only the cosine function is positive in Quadrant IV sinθ cosθ tanθ Udf Udf What happens if one should not remember a value of one of these functions for a common angle value beyond the first quadrant? We may use practice of determination of sign (ASTC) and reference angle to mentally calculate sine, cosine and tangent values for quadrants II, III, and IV as the following examples illustrate. 7 Example : Calculate sin 7. Since < <, we know that 7 lies in the third quadrant. Using the ASTC mnemonic, we know 7 sin <. The reference angle for 7 is - the first quadrant common angle with the same denominator. Therefore, 7 sin = sin =. TrigTable 5 September

11 Example : Calculate Example : Calculate Completing the Table 5 tan. First, verify that 5 7 cos. Verify that 7 5 lies in the fourth quadrant. Therefore, tan <. The reference angle is, so 5 tan = tan =. 7 lies in Quadrant IV so that cos >. Therefore, 7 cos = cos = Quite frankly, few people have rapid recall of the values in the bottom half of the table. That is, most mathematicians are much more familiar with the values of the sine, cosine, and tangent functions than they are with the cotangent, secant, and cosecant functions. However, every mathematician can readily compute the values given their knowledge of the top half of the table. This is because each value in the lower half of the table is a reciprocal of a corresponding value in the upper half of the table. The only challenge is to occasionally rationalize a denominator. Computation is reduced to using function definition, determination of sign by identification of quadrant, identification of reference angle, and computation of the sine, cosine, or tangent of the reference angle values. This is only one more step than what was required for the upper half of the table. The following example illustrates these principles. Example : Calculate sec. Note that lies in the second quadrant and that the secant function is the reciprocal of the cosine function. Therefore, the secant function and the cosine function have the same sign in Quadrant II. By ASTC, the cosine function is negative in the second quadrant. Therefore, sec <. Since is the reference angle, we have sec = sec =. If the students fails to remember that sec =, he or she will remember that sec = = =. cos. TrigTable 5 September