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Lesson 4.7 Inverse trigonometric functions Questions to answer I. What do we know about inverse functions? II. Who are the inverse trigonometric functions? III. How do we evaluate inverse trigonometric function without a calculator? IV. How do we compose trigonometric functions with inverse trig functions? I. we know: 1. 2. 3. 4. functions and their inverses are symmetric by y = x line their composition is equal to x f -1(f(x)) = x, and f(f-1(x)) = x domain of f(x) is the range of f-1 (x) and range of f(x) is the domain of f-1 (x) the function has to be one to one ( pass the horizontal line test) II. Trigonometric functions do not pass Horizontal line test, so they do not have an inverse unless we restrict the domain 1.graph sin(x) on window - л ≤ х ≤ л and л/2 ≤ у ≤ л/2 and see on what interval the function pass the horizontal line test 2.graph the inverse of sine function: sin-1(x) also called arcsine of x and sketch the graph on your notes Sin (x) domain Sin -1(x) or arcsin (x) [- л/2,л/2] [-1 , 1 ] Range [-1 , 1 ] [- л/2 ,л/2] Graph cos(x) and cos -1 (x) find the restricted domain and range and sketch the graph Window – 1.3< x < л and – 1.3< y < л cos (x) cos -1(x) or arccos (x) domain [o, л] [-1 , 1 ] Range [-1 , 1 ] [o, л] Graph tan(x) and tan-1(x) , find restricted domain and sketch the graph, standard window. tan (x) tan -1(x) or arctan (x) All real domain [- л/2, л/2] Range All real [- л/2 , л/2] III how do I solve without calculator. (and with calculator) 1. sin – 1 (1/2 ) 7. cos – 1 (1 ) 2. sin 3 -1 2 8. cos – 1(cos(- 1.1 )) 3. sin – 1 (л /2 ) 9. 4. sin – 1(sin л /9 ) 10. sin( arctan ( 2 )) sin – 1(sin 5л / 6) tan – 1 ( 3 ) 11. tan (arcsin (- 5/7 ) 5. 6. cos – 1 2 2 12. tan (cos -1 (0.28)) Answers When you solve it is important to remember where is the inverse function interval on the unit circle 1. sin – 1 (1/2 ) = x then sin x = ½ that means x = л/ 6 or 30º . 2. sin 3 -1 = 2 x then sin x = 3 2 and x= 240° or 300° . we choose 300° (- л /3) because is in the interval (- л /2, л /2) 3. sin – 1 (л /2 ) no solution because domain of arcsin is [ -1, 1] and л /2 >1 4. sin – 1(sin л /9 ) remember composition of function ? f-1(f(x) = x our functions are inverses of each other so the answer is л /9 5. sin – 1(sin 5л / 6) this should be 5л / 6 but is not because it is not in the restricted interval of sin. So we choose the other quadrant where sin has the same value л - 5л/6 = л/6 6. tan – 1 ( 3 ) = x then tan x = ( 3 ) and x = - 60° or – л/3 7. cos – 1 (1 ) = x then cos x = 1 and x is zero 8. cos – 1(cos(- 1.1 )) should be -1.1 however it is not in the restricted interval for cosine so we choose the other value 1.1 IV How do we compose trigonometric functions with inverse trig functions? Composing trig function with arctan Consider a right triangle with one angle , the opposite side x, and adjacent side = 1. 2 1 x Hypotenuse should be Then tan θ = x and tan -1(x) = θ x Sin (tan -1x) = 1 x2 1 Cos (tan -1x) = 1 x 2 and so on