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# 4.1.90001 - 4.1 Exercises 219 Since sin2(0 = 1 —...

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Unformatted text preview: 4.1 Exercises 219 Since sin2(0) = 1 — 0082(6),W6 have 1 _ x2 + 1 _ 1 x2 x2+l x2+l .x2+1 x2+l 2 _ _ + l x _ x s1n(6) _ x2 + 1 — x2 + 1. We eliminated the radical and the : sign in the last equation because the sign of x is the same as the sign of sin(0). Specifically, if x > 0, then 0 < 0 < % and sin(6) > 0. Ifx < 0, then —1 < 6 < Oand sin(0) < 0. So sin2(0) = 1 and 2 . x Sin(arctan(x)) = x2 + l >TRY THIS. Find an equivalent algebraic expression for sin(arccot(x) ). l R thought. . . True or False? Explain. - —1 __ - - _ —1 _ . s1n (O) — s1n(0) 2. s1n(37T/4) — 14W 8. sec(sec (2)) — 2 3. cos—1(0) =' 1 4. sin—1(V2/2) = 135° 9. The functions f(x) = sin—1x and f‘1(x) = sin xarein- 1 1 verse functions. 5. cot’ (5) = t _1 5 6. sec—1(5) = cosT1(‘0.2) an ( ) 10. The secant and cosecant functions are inverses of each 7. sin(cosrl(\/§/2)) : 1 Ni ‘ other. ‘ ’XERCISES 4.1 Fill in the blank. I, , Find the exact value of each expression in degrees without using a « 7 calculator or table. 1. The _+ ofy = arcsin(x) is [—l, l]. , . " 13. arcsin(—1) ‘ 14. sin’1(l/\/2) ‘ 2. The __ of y = arccos(x) is [0, 7r]. ’ i 15. cos_1(—\/2/2) 16. cos—1(\/§/2) 3. The — ofy = arctan(x) is (—00, 00). , 17. arcsin(0.5) 18. sin‘1(\/§/2) 4. The—ofy = tan“1(x) is (—77/2, 7r/2). 19. arccos(-—l) 20. arccos(0) Find the exact value of each expression without using a calculator or table. "’ Find the exact value of each expression without using a calculator or table. 5. s1n-1(—1/2) , 6. sin—1(0) 21. tan’1(-—l) 22. cot—1(l/\/§) 7. arcsin(l/2) S1 . 8. arcsin(\/§/2) _ f ' j 23. sec’1(2) 24. csc_1(2/\/§) 9. cos—1(V2/2) ' 1 .._ 10. cos—1(1) ‘ 25. arcsec(\/2) 26. arctan(—l/\/§) 11. arccos(l/2) 12. arccos(—\/§/2) 27. arccsc(—2) 28. Sarccot(—\/§) 29. tan“1(0) ‘ 30. sec—1(1) ...
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