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Math 170 - Cooley Pre-Calculus OCC Section 5.5 – The Inverse Trigonometric Functions We will now construct inverse functions from our six trigonometric functions by restricting the domain of the each function so that it will be one-to-one. We need each graph of a function and its inverse to be able to pass the Vertical Line Test (to be a function) and the Horizontal Line Test (to be one-to-one which means invertible.) See section 1.8. y = sin x y = cos x x x y = sin x y = cos x 2 x 2 y = sin–1 x 0 x y = cos–1 x y = tan x x , except for odd multiples of 2 y = tan x 2 x 2 y = tan–1 x -1- Math 170 - Cooley Pre-Calculus OCC Section 5.5 – The Inverse Trigonometric Functions Inverse Function Name Usual Notation Definition Domain Range Inverse Sine (arcsine) y sin 1 x y = arcsin x x = sin y [–1, +1] 2 , 2 Inverse Cosine (arccosine) y cos 1 x y = arcos x x = cos y [–1, +1] 0, Inverse Tangent (arctangent) y tan 1 x y = arctan x x = tan y (–, +) 2 , 2 Inverse Function Name Usual Notation Inverse Cosecant (arccosecant) y csc 1 x y = arccsc x y sec 1 x Inverse Secant (arcsecant) y = arcsec x Inverse Cotangent (arccotangent) Definition Domain x = csc y (–, –1] [+1, +) x = sec y (–, –1] [+1, +) x = cot y (–, +) 1 y cot x y = arccot x Range 2 y 2 , y0 0 y , y 2 0, Properties of Inverse Functions sin 1 sin x x where x cos1 (cos x) x 2 2 1 sin sin x x where 1 x 1 cos cos x x 1 tan 1 tan x x tan tan x x 1 where x where 0 x where 1 x 1 2 2 where x Notation for Inverse Functions The inverse functions are denoted by a superscript “–1” between the name of the function and its argument. This notation does not mean an exponent of –1, but is similar to inverse function notation such as f 1 read “f inverse”. So, sin 1 x is read “sine inverse of x” , and … sin x 2 sin 2 x , but 1 sin 1 x sin x . (Note: sin x 1 1 csc x , so obviously sin 1 x csc x or we wouldn’t have different names for them!) sin x -2- Math 170 - Cooley Pre-Calculus OCC Section 5.5 – The Inverse Trigonometric Functions 1 Example: Find the exact value of the expression: sin cos1 2 Solution So, we need to find the angle , 0 , whose cosine equals cos 1 . Thus, 2 1 , 0 2 3 1 3 Now, sin cos1 sin 2 3 2 Example: Use a calculator to find the value of the following expression rounded to two decimal places. csc 1 (5) Solution 1 1 , since csc1 (5) sin 1 . 2 2 5 5 1 1 Now, sin , so lies in quadrant I. The calculator yields sin 1 0.20 , which is an 5 5 So, we seek to find the angle , , whose sine equals angle in quadrant I, so csc1 (5) 0.20 Example: Use a calculator to find the value of the following expression rounded to two decimal places. cot 1 ( 5) Solution So, we seek to find the angle , 0 , whose tangent equals 1 , since 5 1 1 cot 1 ( 5) tan 1 . Now, tan 5 , so lies in quadrant II. The calculator yields 5 1 tan 1 0.42 , which is an angle in quadrant IV. Since, is in quadrant II, 5 0.42 2.72 . Therefore, cot 1 ( 5) 2.72 . -3- Math 170 - Cooley Pre-Calculus OCC Section 5.5 – The Inverse Trigonometric Functions Exercises: Find the exact value of each expression without using a calculator. 1) cos 1 1 2) arccos(1) 3) tan 1 (1) 3 4) sin 1 2 3 5) tan 1 3 2 6) csc 1 3 7) sec1 (1) 8) arccsc(1) 9) cot 1 1 10) cot 1 0 Use a calculator to find the value of each expression rounded to two decimal places. 11) cos 1 0.6 12) sin 1 (0.23) 13) tan 1 (0.4) 14) cos1 (0.44) 15) sec1 ( 3) 16) cot 1 ( 8.1) Find the exact value, if any, of each expression. Do not use a calculator. 17) sin 1 sin 10 2 18) cos cos 1 3 3 19) sin 1 sin 7 21) tan tan 1 2 7 22) cos 1 cos 4 23) cos1 cos 20) sin sin 1 2 24) cos cos 1 -4- Math 170 - Cooley Pre-Calculus OCC Section 5.5 – The Inverse Trigonometric Functions Exercises: Find the exact value of each expression. 1 25) tan sin 1 2 3 26) cos sin 1 2 27) sec tan 1 3 2 28) tan 1 tan 3 2 29) cos sin 1 3 30) sin tan 1 3 1 31) csc tan 1 2 32) sec1 (2) 33) csc1 Answers: 1) 0; 2) ; 3) 4 ; 4) 3 ; 5) 6 ; 6) 3 ; 7) 0 ; 8) 2 ; 9) 4 ; 10) 2 2 ; 11) 0.93; 2 3 ; 18) ; 19) ; 3 10 7 3 1 20) Not defined; 21) –2; 22) ; 23) ; 24) Not defined; 25) ; 26) ; 27) 2; 28) ; 3 4 3 2 7 3 10 3 2 29) ; 30) ; 31) ; 32) ; 33) . 3 10 3 3 4 12) 0.288 0.23 ; 13) –0.38; 14) 2.03; 15) 1.91; 16) 3.02; 17) -5-