Inverse Trigonometric Functions Properties and Formulae ANIL SHARMA K.V. HIRA NAGAR

we have studied that the inverse of a function f, denoted by f –1, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inversesThe concepts of inverse trigonometric functions is also used in science and engineering we have studied that the inverse of a function f, denoted by f –1, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inverses.

Pretesting Questions:- Q 1) Can you suggest the restricted Domains of each T ratios separately at which the functions are one and onto ? Q2) When a function is said to be invertible. ? Q3 Evaluate the followings a) Sin -1 (1/2) b) Sin -1 (-1/2) c) Tan -1 (1) d) Sin -1 (2sin∏/6) Q4) If Sin -1 ( x-1) =∏/4 then find the value of x.

Inverse Trigonometric function – Properties cos -1 (-x) =  - cos -1 x if x is in [-1,1] sin -1 (-x) = -sin -1 (x) if x is in [-1,1] tan -1 (-x) = - tan -1 x if x is in ( - ,  ) cot -1 (-x) =  - cot -1 x if x is in (- ,  ) cosec -1 (-x) = - cosec -1 x if x is in (- ,-1] U [1,  ) sec -1 (-x) =  - sec -1 x if x is in (- ,-1] U [1,  )

Other important properties If x > 0, y > 0 and xy < 1 If x > 0, y > 0 and xy > 1 If x<0,y<0 and xy < 1 sin -1 x+ cos -1 x =  /2 ; if x is in [-1,1]

Find the value of Solution :

Other important properties sin -1 x+ cos -1 x =  /2 ; if x is in [-1,1] If x > 0, y > 0 and xy < 1 If x<0,y<0 and xy < 1 If x > 0, y > 0 and xy > 1

Prove that Solution : And L.H.S. of the given identity is  + 

In triangle ABC if A = tan -1 2 and B = tan -1 3, prove that C = 45 0 Solution : For triangle ABC, A+B+C = 

Inverse Trigonometric function – Conversion To convert one inverse function to other inverse function : 1.Assume given inverse function as some angle ( say  ) 2. Draw a right angled triangle satisfying the angle. Find the third un known side 3.Find the trigonometric function from the triangle in step 2. Take its inverse and we will get  = desired inverse function

The value of cot cosec -1  5 is (a)  /3 (b)  /2 ( c)  /4 (d) none Step 1 Assume given inverse function as some angle ( say  ) Let cot cosec -1  5 = x + y, Where x = cot -1 3 ; cot x = 3 and y = cosec -1  5 ; cosec y =  5

If sin -1 x + sin -1 (1- x) = cos -1 x, the value of x could be (a) 1, 0 (b) 1,1/2 (c) 0,1/2 (d) 1, -1/2 Solution :

If cos -1 x + cos -1 y + cos -1 z = , Then prove that x 2 +y 2 +z 2 = 1 - 2xyz Solution : and given : A+B+C =  Now, L.H.S. = cos 2 A + cos 2 B +cos 2 C = cos 2 A + 1- sin 2 B +cos 2 C = 1+(cos 2 A - sin 2 B) +cos 2 C