Download 2 , arccos x or cos sin−1 x= y⇔sin y=x , − 2 2 cos

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