1st Year Math Notes Chapter 13
Important Complete 1st Year Math Notes Chapter 13 written by Professor M. Sulman Sherazi Suib. These notes are very helpful in the preparation of Mathematics for first years students and these are according to the paper patterns of all Colleges in the World.
Summary and Contents:
Topics which are discussed in the notes are given below:
- Our comprehensive 1st Year Math Notes Chapter 13 will ensure you're fully prepared for your exams.
- Definition of a domain of a function
- Definition range of function
- Graph of the sine function
- Graph of the cosine function
- Graph of the tangent function
- Graph of the cotangent function
- Graph of the cosecant function
- Graph of the secant function
- Definition of function with examples
- Definition of into function with examples
- Definition of objective function with examples
- Definition of injective function with examples
- Definition of bijective function with examples
- Inverse trigonometric functions
- All solved exercises of chapter Number. 13: Inverse Trigonometric Functions Class 11
- The inverse sine function
- Principle and function
- inverse principle Sine function
- Properties of inverse principle sine function
- The inverse Cosine function
- The inverse Tangent function
- The inverse Cotangent function
- The inverse Cosecant function
- The inverse Secant function
- We have been inding the values of trigonometric functions for given measures of the angles. But in the application of trigonometry, the problem has also been the other way round and we are required to ind the measure of the angle when the value of its trigonometric function is given. For this purpose, we need to have the knowledge of inverse trigonometric functions. In chapter 2, we have discussed inverse functions. We learned that only a one-to- one function will have an inverse. If a function is not one-to-one, it may be possible to restrict its domain to make it one-to-one so that its inverse can be found. In this section we shall deine the inverse trigonometric functions.
- We observe that every horizontal line between the lines y = 1 and y = -1 intersects the graph ininitly 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 igure 5. This inverse function is called the inverse cosine function and is written as cos-1x or arccosx. The Inverse Cosine Function is deined 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 relecting the restricted portion of the graph of y = cos x about the line y = x as shown in igure 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 .
- Inverse Cotangent, Secant and Cosecant Functions These inverse functions are not used frequently and most of the calculators do not even have keys for evaluating them. However, we list their deinitions as below: i) Inverse Cotangent function: y = cotx, where 0 7x7p is called the Principal Cotangent Function, which is one-to-one and has an inverse. The inverse cotangent function is deined by: y = cot-1x , if and only if x = coty Where 0 and - < < ∞ < < +∞ y x p The students should draw the graph of y = cot-1 x by taking the relection of y = cotx in the line y = x. This is left as an exercise for them. ii) Inverse Secant function y = sec x, where 0 and p≤≤ ≠ p is called the Principal Secant Function, which is one-to-one and has an inverse. The Inverse Secant Function is deined by: y = sec-1x. if and only if x = secy