# 4.1.60001 - 216 Chapter 4 Solving Conditional Trigonometric...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 216 Chapter 4 Solving Conditional Trigonometric Equations Domain (-oo, -1] U [1. 00) Range Pg, 0) L406] Identities for 030“, sec", and cot‘1 Domain (-00,-1]U[1,00) ‘ Domain (—00,00) Range [OH-72!)U(§77] Range (0,77) When studying inverse trigonometric functions, you should first learn to ev ‘ ate sin’I, cosﬁl, and tan—1. Those values can be used along with the ident csc a = l/sin oz, see a = l/cos a, and cota = 1/tanoz to evaluate csc_1, sec and cotgl. For example, sin(77/6) = 1/2 and csc(77/6) = 2. So the angle whose}; cosecant is 2 is the same as the angle whose sine is l / 2. In symbols, 1 77 —1 = ' —1 — = — csc (2) sm < 2) 6 . 1 111 general, 080—196 =Vsin_1(1/x). Likewise, sec’ x = cos’1(1/x). For the inverse: cotangent, cot—ix = tan—1(1/x) only for positive values of x, because of the choi of (0, 77) as the range of the inverse cotangent. We have cot—1(0) = 77/2 and if x is; negative cot—1(x) = tan’1(l/x) + 77. i Another relationship between cot_1 and tan_1 can be obtained from their graphs : in the Function Gallery on pages 215—216. Since the graph of y = cot—1(x)rcan be ob- :I tained from the graph of y = tan—1(x) by reﬂecting with respect to the x-axis and? translating upward a distance of 77/2, we have cot—1(x) = i—tan_1(x) + 7/2 or (x) = 77/2 — tan“1(x). These identities are summarized below. g) V /x)for|x|‘: .1, ;_ tan—1(l/x) _ i ’- fory)x«>,; ,0 , i: tanﬂ(’l/x)-l- n77 I‘forjé <70 ,_77/2,:' T forx:0 _‘ - EXAMPLE 7 Evaluating the inverse functions Find the exact value of each expression without using a table or a calculator. a. arcsec(—2) b. csc‘1(\/2) c. arccot(—1/\/§) Solution 3. To evaluate the inverse secant we use the identity sec—1(x) = cos—1(1/x). In this case, the arc whose secant is —2 is the same as the arc whose cosine is — 1 / 2. So we must find cos—1(—-1/2). Since cos(277/3) = —l/2 and since 277/3 is in the range of arccos, we have arccos(—1/2) = 277/3. So 2 arcsec(—2) = arccos(—1/2) = 771-. ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern