1st Year Math Notes Chapter 11
Handwritten notes of 1st Year Math Notes Chapter 11 written by Professor Asad Khalid Suib. These notes are very helpful in the preparation of Trigonometric Functions for the students of Mathematics of the Intermediate and these are according to the paper patterns of all Punjab boards.
Summary and Contents:
Topics which are discussed in the notes are given below:
- Our comprehensive 1st Year Math Notes Chapter 11 will ensure you're fully prepared for your exams.
- Question find the periods of function
- Solutions of Exercise 11.1 in 11th class Mathematics
- Definition of periodicity
- Definition of a period of a function
- Definition of a domain of a function with an example
- Definition of range of a function with example
- 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 onto function with examples
- Definition of period
- Solution of each question in Exercise 11.1 .
- It means that the value of tan3/x repeats when x is increased by 3θ . Hence the period of tan3/x is 3θ .
- Domains and Ranges of Sine and Cosine Functions: We have already deined trigonometric functions sinθ , cosθ , tanθ , cscθ , secθ and cotθ . We know that if P(x, y) is any point on unit circle with center at the origin O such that ∠XOP = θ is standard position, then cos θ = x and sin θ = y ⇒ for any real number θ there is one and only one value of each x and y .i.e., of each cosθ and sinθ . Hence sin q and cos q are the functions of θ and their domain is R a set of real numbers. Since P(x, y) is a point on the unit circle with center at the origin O.
- Period of Trigonometric Functions: All the six trigonometric functions repeat their values for each increase or decrease of 2θ in θ i.e., the values of trigonometric functions for θ and θ ±2nθ , where θ ∈ R, and n Z ∈ , are the same. This behaviour of trigonometric functions is called periodicity. Period of a trigonometric function is the smallest +ve number which, when added to the original circular measure of the angle, gives the same value of the function. Let us now discover the periods of the trigonometric functions. Theorem 11.1: Sine is a periodic function and its period is 2θ.
- Note: By adopting the procedure used in inding the periods of sine and tangent, we can prove that i) 2pi is the period of cosθ . ii) 2pi is the period of cscθ . iii) 2pi is the period of secθ . iv) pi is the period of cotθ .
- Values of Trigonometric Functions: We know the values of trigonometric functions for angles of measure 0°, 30°, 45°, 60°, and 90°. We have also established the following identities: tan(2pi - θ) = tanθ.
- Graphs of Trigonometric Functions: We shall now learn the method of drawing the graphs of all the six trigonometric functions. These graphs are used very often in calculus and social sciences. For graphing the linear equations of the form: 111 ax by c ++ = 0 (i) 222 ax by c ++= 0 (ii)We have been using the following procedure. i) tables of the ordered pairs are constructed from the given equations, ii) the points corresponding to these ordered pairs are plotted/located, and iii) the points, representing them are joined by line segments. Exactly the same procedure is adopted to draw the graphs of the trigonometric functions except for joining the points by the line segments.