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7 – 5 Derivatives of Inverse Trigonometric Functions Since trigonometric functions are periodic, they are not 1 – 1. They do not pass the horizontal line test. To make trigonometric functions 1- 1 we need to restrict the domain. If we restrict the domain of sin x to the interval defined by − , 2 2 then its inverse, arcsin x or sin −1 x is also a function, making sin x a 1 – 1 function. For the cosine function we need to do something slightly different, as follows: If we restrict the domain of cos x to the interval defined by 0, then its inverse, arccos x or cos−1 x is also a function, making cos x a 1 – 1 function. With the above limits to the domains of the sine and cosine function, we can conclude: ≤ y≤ 2 2 −1 cos x = y ⇔ cos y = x , 0≤ y ≤ −1 sin x = y ⇔ sin y = x , − This also means that sin−1 sin x =x , − ≤ x ≤ 2 2 −1 sin sin x =x , −1≤ x ≤1 cos−1 cos x = x , 0≤ x ≤ cos cos−1 x = x , −1≤ x ≤1 For the tangent function, the following restriction makes it 1 – 1 tan−1 x = y ⇔ tan y = x , − ≤ y≤ 2 2 Study the examples in the book. Assignment: page 334 # 1 d, e, f, 2 d, e, f, 3 d, e, f, 4 d, e, f, 5 a, b Assignment: page 339 – 340 # 1 a - e, 2, 4, 7 b