Tiêu đề 2. INVERSE TRIGNOMETRIC FUNCTIONS.pdf ✅

Revision Notes As we have learnt in class XI, the domain and range of trigonometric functions are given below: S. No. Function Domain Range (i) sine  [– 1, 1] (ii) cosine  [– 1, 1] (iii) tangent − = { } x x: (2 1 n n + ∈ ) ; 2 π  (iv) cosecant  – {x : x = np, n Î Z}  – (– 1, 1) (v) secant  R x − = { } : ( x n2 1 + ∈ ) ;n 2 π   – (– 1, 1) (vi) cotangent  – {x : x = np, n Î Z}  1. Inverse function We know that if function f : X ® Y such that y = f(x) is one-one and onto, then we define another function g : Y ® X such that x = g(y), where x Î X and y Î Y, which is also one-one and onto. In such a case, Domain of g = Range of f and Range of g = Domain of f g is called the inverse of f g = f –1 or Inverse of g = g –1 = (f –1) –1 = f The graph of sine function is shown here: Y X' X Y' 5 –2 2 2 –1 2 2 3 1 2 3 0 2 25 y x =sin Principal value of branch function sin–1:It is a function with domain [– 1, 1] and range − −      −      3 2 2 2 2 π π π π ,,, or π π 2 3 2 ,       and so on corresponding to each interval, we get a branch of the function sin– 1 x. The branch with range −      π π 2 2 , is called the principal value branch. Thus, sin–1 : [–1, 1] ® −      π π 2 2 , . [SQP 2023-2024] UNIT-I : RELATIONS AND FUNCTIONS INVERSE TRIGONOMETRIC FUNCTIONS 2 CHAPTER C H A P T E R 1 Learning Objectives After going through this chapter, the students will be able to learn:  Domain and Range of Inverse Trigonometric Functions  Principal Branch value of Inverse Trigonometric Functions  Graphs of Inverse Trigonometric Functions
MATHEMATICS, Class-XII Y' X' X 3 2 3 Y 5 / 2 3 / 2 /2 1 /2 – 3/2 2 – 5 / 2 y x = sin–1 0 1 Principal value branch of function cos–1: The graph of the function cos–1 is as shown in figure. Domain of the function cos–1 is [–1, 1]. Its range in one of the intervals (– p, 0), (0, p), (p, 2p), etc. is one-one and onto with the range [– 1, 1]. The branch with range (0, p) is called the principal value branch of the function cos–1. Thus, cos–1 : [– 1, 1] ® [0, p] 5 —2 • • • 2 3 —2 —2 1 X' X –1 —2 – 3 —2 – –2 5 —2 – Y' Y – y x = cos–1 0 [Board, 2023] 1 —2 5 —2 X 3 —2 2 –1 —2 – X' 5 —2 – 3 —2 – –2 – Y Y' • • • y x = cos 0 Principal value branch function tan–1: The function tan–1 is defined whose domain is set of real numbers and range is one of the intervals,  − −       −            3 2 2 2 2 2 3 2 π π π π π π , , , , , ,..... Graph of the function is as shown in the figure: The branch with range −       π π 2 2 , is called the principal value branch of function tan–1. Thus, tan : , − →  −      1 2 2  π π . [Board -2023] Principal value branch of function cosec–1 : The graph of function cosec–1 is shown in the figure. The cosec–1 is defined on a function whose domain is  – (– 1, 1) and the range is any one of the interval,

MATHEMATICS, Class-XII Inverse Function Domain Principal Value Branch sin–1 cos–1 cosec–1 sec–1 tan–1 cot–1 [ – 1, 1] [– 1, 1]  – (– 1, 1)  – (– 1, 1)   −      π π 2 2, [0, p] −      − π π 2 2, {0} 0 2 , π π [ ] − { }  −      π π 2 2, (0, p) (3) Principal Value: Numerically smallest angle is known as the principal value. Finding the principal value: For finding the principal value, following algorithm can followed : STEP 1: First draw a trigonometric circle and mark the quadrant in which the angle may lie. STEP 2: Select anti-clockwise direction for 1st and 2nd quadrants and clockwise direction for 3rd and 4th quadrants. STEP 3: Find the angles in the first rotation. STEP 4: Select the numerically least (magnitude wise) angle among these two values. The angle thus found will be the principal value. STEP 5: In case, two angles one with positive sign and the other with the negative sign qualify for the numerically least angle then, it is the convention to select the angle with positive sign as principal value. The principal value is never numerically greater than p. (4)To simplify inverse trigonometric expressions, following substitutions can be considered: Expression Substitution a2 + x2 or a x 2 2 + x = a tan q or x = a cot q a2 – x2 or a x 2 2 − x = a sin q or x = a cos q x2 – a2 or x a 2 2 − x = a sec q or x = a cosec q a x a x − + or a x a x + − x = a cos 2q a x a x 2 2 2 2 − + or a x a x 2 2 2 2 + − x2 = a2 cos 2q x a x − or a x x − x = a sin2 q or x = a cos2 q x a x + or a x x + x = a tan2 q or x = a cot2 q Note the following and keep them in mind:  The symbol sin–1 x is used to denote the smallest an- gle whether positive or negative, such that the sine of this angle will give us x. Similarly cos–1 x, tan–1 x, cosec–1 x, sec–1 x and cot–1 x are defined.  You should note that sin–1 x can be written as x. Similarly, other Inverse Trigonometric Functions can also be written as arccos x, arctan x, arcsec x etc.  Keep in mind that these inverse trigonometric re- lations are true only in their domains i.e., they are valid only for some values of ‘x’ for which inverse trigonometric functions are well-defined. MNEMONICS Inverse trigonometric ratio can be used to find the angle of a right triangle when given two sides of the triangle. SOH θ = − sin 1 opposite hypotenuse CAH θ = − cos 1 adjacent hypotenuse TOA θ = − tan 1 opposite adjacent Hypotenuse Adjacent Opposite Example-1 Find the principal value of sin–1 1 2       . Sol. Step 1 : Let y = sin–1 1 2       Þ sin y = 1 2 Step 2 : sin y = 1 2 Þ sin y = sin p 4 Þ y = p 4 Step 3 : We know that the range of the principal value branch of sin−1x is −       p 2 2 , π . Therefore, y = p 4 . Hence, the principal value of sin−1 1 2       is p 4 . The principal value branch of trigonometric inverse functions is as follows: