Page 5 - 11-Math-13 Invnerse Trignometric Functions

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We observe that every horizontal line between the lines y = 1 and y = -1 intersects the

graph infinitly many times. It follows that the cosine function is not one-to-one. However, if

we restrict the domain of y = cosx to the interval [0, p], then the restricted function y = cosx,
0 7x7p is called the principal cosine function; which is now one-to-one and hence will

have an inverse as shown in figure 5.

This inverse function is called the inverse cosine function and is written as cos-1x or arc
cosx.
The Inverse Cosine Function is defined by:

y = cos-1x, if and only if x= cos y.

where 0 7y7p and -17x 7 1.
Here y is the angle whose cosine is x . The domain of the function y = cos-1x is -17x 71
and its range is 07y7p .

The graph of y = cos-1x is obtained by reflecting the restricted portion of the graph of
y = cos x about the line y = x as shown in figure 6.
We notice that the graph of y = cos x is along the x - axis whereas the graph of y = cos-1x
is along the y - axis .

Note: It must be remembered that cos-1x ≠(cosx)-1

Example 2: Find the value of (i) cos-11 (ii) cos-1(- 1 )
2

Solution: (i) We want to find the angle y whose cosine is 1

⇒=cos y 1, 0≤ y≤p

⇒ y =0

∴ cos-11 =0

(ii) We want to find the angle y whose cosine is - 1
2

⇒ cos y =- 1 , 0 ≤ y ≤ p
2

version: 1.1

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