4.4.3 The inverse cosine function and its

. . The inverse cosine function and its properties The cosine function is not one-to-one in the entire domain  . However, the cosine function is one-to-one on the restricted domain , π ] and still, on this restricted domain, the range is − ] Now, let us define the inverse cosine function with [ , ] - as its domain and with , π ] as its range.

Definition . For −≤ , define cos - x as the unique number y in , π ] such that cos y . In other words, the inverse cosine function cos : ] π is defined by cos ( ) x y if and only if cos y ] , π . Note (i) The sine function is non-negative on the interval , π ] , the range of cos - x .

This observation is very important for some of the trigonometric substitutions in Integral Calculus. (ii) Whenever we talk about the inverse cosine function, we have cos : ] π . (iii) We can also restrict the domain of the cosine function to any one of the intervals   ] where it is one-to-one and its range is [ - , ]. The restricted domain , π