CHAPTER C: Inverse trigonometric functions and secant, cosecant and cotangent Learning objectives After studing this chapter, ou should be able to: work with the inverse trigonometric functions sin, cos and tan and be able to draw their graphs over appropriate restricted domains understand the secant, cosecant and cotangent functions sketch the graphs of the secant, cosecant and cotangent functions know and use the two identities relating to the squares of the secant, cosecant and cotangent functions to prove other identities and to solve equations.. Inverse trigonometric functions In section. ou read that in order for an inverse function f to eist, the function f must be one-one. The sine function, defined over the domain of real numbers, is clearl a man-one function as can be seen from its graph. sin So, at first sight, it would seem that the sine function has no inverse. To overcome this problem the domain of the sine function is restricted so that it becomes a one-one function but still takes all real values in the range sin. You can do this b restricting the domain to, where is in radians. The inverse function of sin,, is written as sin. From the general definition of an inverse function ou can deduce that sin sin and.
C: Inverse trigonometric functions and secant, cosecant and cotangent sin has domain and range sin. The graph of sin, is obtained b reflecting the graph of sin,, in the line. Warning: sin. sin Worked eample. Find the eact values of: (a) sin (b) cos sin (a) Since an eact answer is required, using our calculator set in degree mode, sin 0, and 0 80 0 6 radians. So sin 6, since sin 6 and 6 lies between 6 and. (b) Let sin sin. Using the identit cos +sin, ou have cos 8 9. Since has to lie between and ou can deduce that cos is positive. cos cos(sin ) cos 8 9. Although the following direct results are true sin (sin ) for, sin(sin ) for. Composite functions and inverse functions are considered in chapter. ou must be careful when lies outside the given inequalities. The net worked eample shows ou how to deal with such a case.
C: Inverse trigonometric functions and secant, cosecant and cotangent Worked eample. Given that sin, find the eact value of sin sin. Since does not lie between and ou cannot use the direct result. Instead ou evaluate sin sin sin sin. Let sin sin sin sin. The value of which satisfies sin and lies between and is sin sin. The inverse functions of cos and tan are dealt with in a similar wa. The restricted domain for cos is not the same as that used for sin. The inverse function of cos, 0, is written as cos. cos cos and 0. Restricting the domain of cos to would not give the negative values in the range 0 to. cos has domain and range 0 cos. cos The graph of cos, is obtained b reflecting the graph of cos, 0, in the line. Warning: cos. co s 0 cos cos (cos ) for 0, cos(cos ) for. The tangent function can be made one-one b restricting its domain to. The inverse function of tan,, is written as tan. tan tan and.
C: Inverse trigonometric functions and secant, cosecant and cotangent 5 tan has domain all real numbers and range tan. The graph of tan, is obtained b reflecting the graph of tan,, in the line. tan tan Warning: tan. ta n The lines and are vertical asmptotes of the graph of tan. When reflected in the line the become horizontal. The lines and are horizontal asmptotes of the graph of tan. Worked eample. B considering the graphs of 0.5 and cos determine the number of real roots of the equation cos. The equation cos can be rearranged into the form cos 0.5. The number of real roots of the equation correspond to the number of times the graphs cos and 0.5 intersect. The graphs intersect in just one point so the equation cos has onl one real root. cos 0 0.5
6 C: Inverse trigonometric functions and secant, cosecant and cotangent EXERCISE A In this eercise ou ma assume the results in the table opposite. You do not need to learn them for the eam. Find the eact values of: (a) cos (b) cos(cos 0) (c) cos cos (d) cos cos (e) cos sin (f) cos sin Find the eact values of: (a) tan (c) tan tan (b) tan[tan ()] (d) tan cos (e) tan cos (f) sin tan You are given that h() sin cos tan. Find the value of: (a) h(), (b) h(). Determine the number of real roots of the equation sin. 5 Show that the equation tan k, where k is a constant, has onl one real root when k 0 and state its value. 6 For ver large positive values of the constant k, state the number of real roots of the equation k tan. Angle sin cos tan in radians 0 0 0 6 0 0 0. Secant, cosecant and cotangent So far ou have worked with three trigonometric ratios, sine, cosine and tangent. The remaining three trigonometric ratios are secant, cosecant and cotangent. The are written as sec, cosec and cot, respectivel, and are defined as follows: sec co s cosec sin cot c os ta n sin Calculators do not have the function kes for secant, cosecant and cotangent. The net worked eample shows ou how to find the values of these functions. Secant is the reciprocal of cosine. Do not write sec as cos, which means something entirel different as ou saw in section.. Similarl cosec sin and cot tan. sin Used tan c os, from C section 6.5.
C: Inverse trigonometric functions and secant, cosecant and cotangent 7 Worked eample. (a) Find the values of: (i) sec 0, (ii) cosec 5, (iii) cot 50. (b) Given that tan 6, find the eact value of cot 5. 6 (a) (i) sec 0.05 (to d.p.), cos 0 0.766 0 (ii) cosec 5 sin sin 0.68 8 5.70 (to d.p.). 0.587 785 Set our calculator in radian mode. (iii) cot 50 0.6 (to d.p.); tan 50.77 7 (b) cot 5 6 tan 5 tan tan 6 6 6 Used tan ( ) tan. The net two worked eamples show ou how to solve some basic trigonometric equations involving a single secant, cosecant or cotangent function. Worked eample.5 Solve these equations for 0 60. Give our answers to one decimal place. (a) sec.9, (b) cosec.5. (a) sec.9 cos 0.56 5.9 Cos is positive so answers lie in the intervals 0 90 and 70 60. The acute angle cos (0.56 5 ) 58., so 58. or 60 58. To one decimal place, 58., 0.8. 0.5 cos 90 80 70 60
8 C: Inverse trigonometric functions and secant, cosecant and cotangent (b) cosec.5 sin 0.8. 5 Sin is negative so answers lie in the interval 80 60. The acute angle, sin 0.8 5., so 80 5. or 60 5.. To one decimal place,., 06.9. 0.8 sin 90 80 70 60 Worked eample.6 Given that tan, solve the equation cot giving our answer, in terms of, in the interval for 0. cot tan. Since 0, ou require all values for between 0 and. tan (), and tan is periodic with period, tan θ θ or or or 7,, and 5. 8 8 8 8 The net Worked eample shows ou how to use the definitions of secant, cosecant and cotangent to obtain other trigonometric identities. Worked eample.7 Prove the identit cot A sec A cosec A. cot A sec A tan A co s A c os A sin A co s A cosec A sin A sin A tan A c os A EXERCISE B Find the values of: (a) sec 70 (b) cosec 70 (c) cot 0 (d) sec(70 ) (e) cosec 90 (f) cot 0 (g) (h) s ec 60 6 co t 5
C: Inverse trigonometric functions and secant, cosecant and cotangent 9 Find the values of: (a) sec (b) cosec 0.7 (c) cot 0.5 (d) sec() The angles are measured in radians in this question. (e) cosec 8 (f) cot 8 (g) (h) Using the table of results opposite and below, find the eact values of: (a) sec 60 (b) cosec 60 (c) cot 0 (d) sec(80 ) (e) cosec 5 (f) cot 0 (g) sec 0 sec 0 (h) 6 cot 5 7 cot 50 Using the table of results opposite, find the eact values of: (a) cosec (b) cot (c) sec 6 Angle Angle sin cos tan in in degrees radians 0 6 5 60 (d) cot 6 80 0 0 5 Solve these equations for 0 60. Give our answers to one decimal place. (a) sec.8 (b) cosec.5 (c) cot (d) sec. (e) cosec (f) cot. (g) sec 7 (h) 5 cot 6 Solve these equations for 0, giving our answers in radians to three significant figures. (a) sec (b) cosec (c) cot (d) sec 5 (e) cosec (f) cot (g) sec (h) cot
50 C: Inverse trigonometric functions and secant, cosecant and cotangent 7 Prove the following identities: (a) tan A cosec A sec A (b) sin A cot A cos A cot A (c) (d) tan A cosec A sec A cot A t an A (e) sec A cos A tan A sin A sin A (f) cosec A cot A cos A. Graphs of sec,cosec and cot Graph of sec Hint for part (f): Consider ( cos A) (cosec A cot A). cos 0 sec Comparing the graph of sec with the graph of cos ou can see that sec has maimum points where cos has minimum points and has minimum points where cos has maimum points. Secant, like cosine, is a periodic function with period and so the graph repeats itself ever radians. The domain of sec is all real values (k ) and its range is all real values ecept.
C: Inverse trigonometric functions and secant, cosecant and cotangent 5 Graph of cosec Consider cosec sin cosec sec 0 So a translation of the graph of cosec. cos transforms the graph of sec into So, if f() sec, then f cosec. C, section 5.. sin 0 cosec Comparing the graph of cosec with the graph of sin ou can see that cosec has maimum points where sin has minimum points and has minimum points where sin has maimum points. Cosecant, like sine, is a periodic function with period and so the graph repeats itself ever radians. The domain of cosec is all real values k and its range is all real values ecept.
5 C: Inverse trigonometric functions and secant, cosecant and cotangent Graph of cot Consider cot c sin os sin cos cot tan tan So a reflection in the line transforms the graph of tan into the graph of cot. So, if f() tan, then f cot. C EB Q6(c). 0 0 tan cot Cotangent, like tangent, is a periodic function with period and so the graph repeats itself ever radians. The domain of cot is all real values k and its range is all real values. Worked eample.8 (a) Determine the transformation that maps sec onto sec. (b) State the period, in radians, of the graph sec. (c) Sketch the graph of sec for 0,,. (a) Let f() sec then f() sec. The transformation that maps sec onto sec is a stretch of scale factor in the -direction.
C: Inverse trigonometric functions and secant, cosecant and cotangent 5 (b) The period of sec is radians. (c) The graph of sec for 0,, is 0 EXERCISE C (a) Determine the transformation that maps cosec onto cosec. (b) State the period, in radians, of the graph cosec. (a) Determine the transformation that maps cot onto cot. (b) State the period, in radians, of the graph cot. Sketch the graph of sec for. (a) Sketch the graph of cosec for 80 80, 0, 90. (b) Describe a sequence of transformations that maps the curve sec onto the curve cosec.. Identities involving the squares of secant, cosecant and cotangent In C, section 6., ou used the identit cos sin. In this section ou will be shown two similar identities. If ou divide each term in the identit cos sin b cos ou get c os sin cos c os co s
5 C: Inverse trigonometric functions and secant, cosecant and cotangent tan sec Similarl, if ou divide each term in the identit cos sin b sin, ou get c os sin s in s in sin cot cosec These two identities, along with cos sin, are frequentl used to solve trigonometric equations and to prove other identities. The are not given in the eamination formulae booklet so ou must memorise them and know how to appl them. Worked eample.9 Solve the equation sec tan, giving all solutions in the interval 0 60. To solve the equation sec tan use the identit sec tan to get tan tan. tan tan 0 (tan )(tan ) 0 tan, tan. For tan, tan 7.57. For 0 60, tan 7.57, 80 7.57 For tan, tan () 5. For 0 60, tan 80 5, 60 5 The solutions of the equation sec tan, in the interval 0 60, are 7.6, 5, 5.6, 5. The general strateg when asked to solve trigonometric equations which involve the same angle but with a squared trigonometric function is to tr to write the equation as a quadratic equation in the non-squared trigonometric function. As a starter, tr to get a quadratic equation in the nonsquared trigonometric term, tan here. 90 80 70 60
C: Inverse trigonometric functions and secant, cosecant and cotangent 55 Worked eample.0 Solve the equation cosec 5 cot, giving all solutions in the interval 0, to three significant figures. To solve the equation cosec 5 cot use the identit cosec cot to get cot 5 cot cot cot 0 (cot )(cot ) 0 cot, cot tan, tan Tr to write the equation as a quadratic equation in cot. Used tan. co t For tan, tan 0.5 rads. Set calculator in radian mode. For 0, tan 0.5 rads, ( 0.5) rads. For tan, tan () 0.785 rads. For 0, tan ( 0.785) rads, ( 0.785) rads. The solutions of the equation cosec 5 cot, in the interval 0, are 0.5 c,.6 c,.9 c and 5.50 c, to sf. Worked eample. Prove the identit sec A cosec A (tan A cot A)(tan Acot A). sec A cosec A tan A( cot A) tan A cot A (tan A cot A)(tan A cot A) So sec A cosec A (tan A cot A)(tan Acot A). Difference of two squares. EXERCISE D Prove the identit tan A cot A (sec A cosec A)(sec A cosec A). Prove the identit cot A sin A (cosec A cos A)(cosec A cos A). Prove the identit cosec A cos A cosec A.
56 C: Inverse trigonometric functions and secant, cosecant and cotangent Given that cos sin, show that cos sin and hence find the possible values of cot. 5 Given that 5sec tan 9, find the possible values of sin. 6 Given that sec and tan, show that. 7 Prove the identit (sec Atan A)(tan A sec A) cot A cosec A. cosec Acot A cos A 8 Prove the identit (cosec A cot A) c. os A 9 Eliminate from equations cosec and cot. 0 Eliminate from equations cosec and tan. Solve the equation sec tan, giving all solutions in the interval 0 60. Solve the equation sec 5 tan, giving all solutions in the interval 80 80. Solve the equation cosec cot, giving all solutions in the interval 0 60. Solve the equation tan sec, giving all solutions in the interval 80 80. 5 Solve the equation cot cosec, giving all solutions in the interval 0 60. 6 Solve the equation tan sec 0, giving all solutions in the interval 0 60. 7 Solve the equation cot cosec 7, giving all solutions in the interval 0. 8 Solve the equation tan sec, giving all solutions in the interval 0. 9 Solve the equation cot 5 cosec, giving all solutions in the interval 0. 0 Solve the equation cosec cot, giving all solutions in the interval. Given that sec A tan A, show that sec A. Given that cosec cot, show that cosec.
C: Inverse trigonometric functions and secant, cosecant and cotangent 57 Ke point summar sin has domain and range p sin. The graph of sin, is obtained b reflecting the graph of sin,, in the line. Warning: sin. si n sin (sin ) for, p sin(sin ) for. cos has domain and range p 0 cos. The graph of cos, is obtained b reflecting the graph of cos, 0, in the line. cos Warning: cos. co s 0 cos cos (cos ) for 0, p cos (cos ) for. 5 tan has domain all real numbers and range p5 tan. The graph of tan, is obtained b reflecting the graph of tan,, in the line. tan tan Warning: tan. ta n
58 C: Inverse trigonometric functions and secant, cosecant and cotangent 6 sec co s cosec sin cot c os ta n sin p6 7 p50 cos 0 sec
C: Inverse trigonometric functions and secant, cosecant and cotangent 59 8 p5 sin 0 cosec 9 p5 0 0 tan cot 0 tan sec p5 cot cosec p5
60 C: Inverse trigonometric functions and secant, cosecant and cotangent Test ourself Show that the equation cos sin has onl one real Section. root and state its value to three significant figures. Solve the equation cosec( 0 )., Section. for 0 60. Give our answers to one decimal place. B sketching the graphs of cot and for 0, Section. show that the equation cot has one root and eplain wh this root must be in the interval. What to review Solve the equation tan sec 5, giving all solutions Section. in the interval 0. Test ourself ANSWERS 0.707..6, 58.. 0.7 c,.09 c,.9 c, 5.56 c.