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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, π π π π , , , , or π . A student who has completed precalculus with trigonometry should be able to 6 4 3 2 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. usually integer multiples of 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 1 2005 September 26 VALUES OF THE SIX TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES 0 π π π π 6 4 3 2 2π 3 3π 4 5π 6 π 7π 6 5π 4 4π 3 3π 2 5π 3 7π 4 11π 6 2π 3 2 2 2 1 2 0 − 1 2 − 2 2 − 3 2 −1 − 3 2 − 2 2 − 1 2 0 1 2 − 2 2 − 3 2 −1 − 3 2 − 2 2 − 1 2 0 1 2 2 2 3 2 1 sin θ 0 1 2 2 2 3 2 1 cosθ 1 3 2 2 2 1 2 0 tan θ 0 3 3 1 3 Udf − 3 −1 − 3 3 0 3 3 1 3 Udf − 3 −1 − 3 3 0 cot θ Udf 3 1 3 3 0 − 3 3 −1 − 3 Udf 3 1 3 3 0 − 3 3 −1 − 3 Udf sec θ 1 2 3 3 2 2 Udf −2 − 2 2 3 3 −1 2 3 3 − 2 −2 Udf 2 2 2 3 3 1 csc θ Udf 2 2 2 3 3 1 2 3 3 2 2 Udf −2 − 2 2 3 3 −1 − 2 −2 Udf − − − − − 2 3 3 Udf means Undefined. TrigTable 2 2005 September 26 VALUES OF THE SIX TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES π 0 2π sin θ cosθ tan θ cot θ sec θ csc θ Use copies of this page to practice learning the values of the trigonometric functions. TrigTable 3 2005 September 26 How to Learn the Table of Trigonometric Values The table has 6 rows and 18 columns (the θ = 2π column is a repeat of the θ = 0 column) for a total of 144 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 1/3! (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 144 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 10 different numbers. Radian Values of Common First Quadrant Angles π π π π ; call these the common first quadrant angles. These radian 6 4 3 2 measures correspond to degree values of 0, 30, 45, 60, and 90 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. The first quadrant angles of interest have values of 0, , , , and Note that each first quadrant common angle fraction has π 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 sum of each pair is TrigTable π 2 . That is, 0 + π 2 = π 2 and π 6 + π 3 = π 2 . What about 4 π 4 π 4 in that the values can be paired in such a way so that the ? It can be paired with itself: π 4 + π 4 = π 2 . 2005 September 26 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. 0 π π π π 6 4 3 2 sin θ 0 1 2 2 2 3 2 1 sin θ 0 2 1 2 2 2 3 2 4 2 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 0 and count whole numbers to the value 4, one has everything required to reproduce values of the sine function for the common first quadrant angle values. TrigTable 5 2005 September 26 Values of the Cosine for the Common Angles in the First Quadrant ⎛π ⎞ π We may use the identity cosθ = sin⎜ − θ ⎟ and the symmetry of the first quadrant common angles about to develop the table to ⎝2 ⎠ 4 include values of the cosine function for the first quadrant common angles. Using the symmetry of the first quadrant common angles about π 4 , we see that the cosine values repeat the sine values, but in decreasing order. 0 TrigTable π π π π 6 4 3 2 sin θ 0 1 2 2 2 3 2 1 cosθ 1 3 2 2 2 1 2 0 6 2005 September 26 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: 0 π π π π 6 4 3 2 sin θ 0 1 2 2 2 3 2 1 cosθ 1 3 2 2 2 1 2 0 tan θ 0 3 3 1 Udf 3 It may be instructive to review the arithmetic required for rationalization of the denominator. The arithmetic for tan π 6 is developed: 1 1 2 1 1 3 3 tan = 2 = ⋅ = = ⋅ = 6 3 3 3 3 3 2 3 2 π Note that the three tangent values for tan π = 3 tan π and tan π = 3 tan 4 6 3 permitted (Udf means undefined.) TrigTable π π 6 π 4 , 4 , and π 3 form a geometric sequence with . Note also that π 2 3 as the common ratio; that is, is not in the domain of the tangent function as division by 0 is not 7 2005 September 26 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: 0 π π π π 6 4 3 2 2π 3 3π 4 5π 6 π 7π 6 5π 4 4π 3 3π 2 5π 3 7π 4 11π 6 2π Quadrant Boundaries 3π , and 2π . Note that the sequence of denominators, 6, 4, and 3, repeats within each 2 2 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: The quadrant boundaries appear at 0, 0 π , π, π π π π 6 4 3 2 π 3 4 6 6 4 3 3π 2 2π 3 4 6 Second Quadrant We previously noted the coefficient of pi in the numerator was 1 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. 3π 2π we have 2 = 3 – 1. For we have 3 = 4 – 1, etc. It is a simple matter to complete the second quadrant common That is, for 3 4 angle values. 0 TrigTable π π π π 6 4 3 2 2π 3 3π 4 5π 6 8 π 6 4 3 3π 2 2π 3 4 6 2005 September 26 Third Quadrant The coefficient of pi in the third quadrant is always one more than the value of the denominator. That is, for 7π we have 7 = 6 + 1. It 6 is a simple matter to complete the third quadrant common angle values. 0 π π π π 6 4 3 2 2π 3 3π 4 5π 6 π 7π 6 5π 4 4π 3 3π 2 2π 3 4 6 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 = 2 ⋅ 4 − 1 . This allows us to easily complete the sequence of common angle values. 0 TrigTable π π π π 6 4 3 2 2π 3 3π 4 5π 6 9 π 7π 6 5π 4 4π 3 3π 2 5π 3 7π we have 4 7π 4 11π 6 2π 2005 September 26 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. 0 π π π π 6 4 3 2 sin θ 0 1 2 2 2 3 2 1 cosθ 1 3 2 2 2 1 2 0 tan θ 0 3 3 1 3 Udf 2π 3 3π 4 5π 6 π 7π 6 3 2 2 2 1 2 0 − 1 2 − 2 2 − 3 2 − 3 −1 − 3 3 − 5π 4 4π 3 3π 2 5π 3 7π 4 11π 6 2π 1 2 − 2 2 − 3 2 −1 − 3 2 − 2 2 − 1 2 0 −1 − 3 2 − 2 2 − 1 2 0 1 2 2 2 3 2 1 0 3 3 1 3 Udf − 3 −1 − 3 3 0 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. 7π 7π 3π 7π . Since π < < , we know that lies in the third quadrant. Using the ASTC mnemonic, we know 6 6 6 2 7π π 7π sin < 0 . The reference angle for is - the first quadrant common angle with the same denominator. Therefore, 6 6 6 Example 1: Calculate sin sin TrigTable 7π 1 π = − sin = − . 6 6 2 10 2005 September 26 Example 2: Calculate tan 5π 5π 5π π . First, verify that lies in the fourth quadrant. Therefore, tan < 0 . The reference angle is , so 3 3 3 3 tan Example 3: Calculate cos 5π π = − tan = − 3 . 3 3 7π 7π 7π 2 π 7π = cos = . Verify that lies in Quadrant IV so that cos > 0 . Therefore, cos . 4 4 4 4 2 4 Completing the Table 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. 2π 2π . Note that lies in the second quadrant and that the secant function is the reciprocal of the cosine 3 3 function. Therefore, the secant function and the cosine function have the same sign in Quadrant II. By ASTC, the cosine function is π 2π 2π π negative in the second quadrant. Therefore, sec < 0 . Since is the reference angle, we have sec = − sec = −2 . If the 3 3 3 3 Example 1: Calculate sec students fails to remember that sec π 3 = 2 , he or she will remember that sec TrigTable π 3 = 1 cos π 3 11 = 1 = 2. ⎛1⎞ ⎜ ⎟ ⎝ 2⎠ 2005 September 26