3. EVALUATION OF TRIGONOMETRIC FUNCTIONS


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1 . EVALUATIN F TIGNMETIC FUNCTINS In this section, we obtain values of the trigonometric functions for quadrantal angles, we introduce the idea of reference angles, and we discuss the use of a calculator to evaluate trigonometric functions of general angles. In Definition., the domain of each trigonometric function consists of all angles for which the denominator in the corresponding ratio is not zero. Because r > 0, it follows that sin /r and cos /r are defined for all angles. However, tan / and sec r/ are not defined when the terminal side of lies anlong the ais (so that 0). Likewise, cot / and csc r/ are not defined when the terminal side of lies along the ais (so that 0). Therefore, when ou deal with a trigonometric function of a quadrantal angle, ou must check to be sure that the function is actuall defined for that angle. Eample Find the values (if the are defined) of the si trigonometric functions for the quadrantal angle 90 (or ). In order to use Definition, we begin b choosing an point ( 0, ) with > 0, on the terminal side of the 90 angle (Figure ). Because 0, it follows that tan 90 and sec 90 are undefined. Since > 0, we have r Therefore, sin 90 cos 90 0 r r r 0 csc 90 cot The values of the trigonometric functions for other quadrantal angles are found in a similar manner. The results appear in Table.. Dashes in the table indicate that the function is undefined for that angle. Table. degrees radians sin cos tan cot sec csc
2 It follows from Definition. that the values of each of the si trigonometric functions remain unchanged if the angle is replaced b a coterminal angle. If an angle eceeds one revolution or is negative, ou can change it to a nonnegative coterminal angle that is less than one revolution b adding or subtracting an integer multiple of 0 (or radians). For instance, sin 50 sin( 50 0 ) sin 90. sec 7 sec ( 7 ( )) sec. cos ( 0 ) cos ( 0 + ( 0 ) ) cos 0. In Eamples. and., replace each angle b a nonnegative coterminal angle that is less than on revolution and then find the values of the si trigonometric functions (if the are defined). Eample B dividing 0 b 0, we find that the largest integer multiple of 0 that is less than 0 is Thus, 0 ( 0 ) (r we could have started with 0 and repeatedl subtracted 0 until we obtained 0.) It follows that sin 0 sin 0 cos 0 cos 0 tan 0 tan 0 csc 0 csc 0 sec 0 sec 0 cot 0 cot 0 Eample We repeatedl add to until we obtain a nonnegative coterminal angle: + (still negative) +. Therefore, b Table. for quadrantal angles, sin sin cot cot 0 cos cos 0 csc csc and both tan and sec are undefined. 9
3 Table. degrees radians sin cos tan cot sec csc Figure. (a) (b) (c) (d)
4 Eample Find the reference angle for each angle. (a) 0 (b) (a) B Figure.(a), (c) 0 (d) 0. (b) B Figure.(b),. (c) B Figure.(c), (d) B Figure.(c),.. The value of an trigonometric function of an angle is the same as the value of the function for the reference angle,, ecept possibl for a change of algebraic sign. Thus, sin ± sin, cos ± cos, and so forth. You can alwas determine the correct algebraic sign b considering the quadrant in which lies. Section Problems In problems and, find the values (if the are defined) of the si trigonometric functions of the given quadrantal angles. (Do not use a calculator.). (a) 0 (b) 80 (c) 70 (d) 0. [When ou have finished, compare our answers with the results in Table.]. (a) (b) (c) 7 (d) (e) 7. In Problems to, replace each angle b a nonnegative coterminal angle that is less than one revolution and then find the eact values of the si trigonometric functions (if the are defined) for the angle
5 5. What happens when ou tr to evaluate tan 900 on a calculator? [Tr it.]. Let be a quadrant III angle in standard position and let be its reference angle. Show that the value of an trigonometric function of is the same as the value of, ecept possibl for a change of algebraic sign. epeat for in quadrant IV. In problems 7 to, find the reference angle for each angle, and then find the eact values of the si trigonometric functions of Complete the following tables. (Do not use a calculator.) degrees radians sin cos tan
6 degrees radians cot sec csc A calculator is set in radian mode. is entered and the sine (SIN) ke is pressed. 0 The displa shows. 0. But we know that sin 0. Eplain. In problems 59 to, use a calculator to verif that the equation is true for the indicated value of the angle. sin 59. tan for 5. cos 0. (cos )(tan ) sin for 7. cos + sin for. + tan sec for Verif that for 0, 0, 5, 0, 90, we have sin,,,, and 0 respectivel. [Although there is no theoretical significance to this pattern, people often use it as a memor aid to help recall these values of sin.]
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