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Page 1 4. Inverse Trigonometric Functions Exercise 4.1 1 A. Question Find the principal value of each of the following: Answer Let Then sin y = We know that the principal value of is Therefore the principal value of is 1 B. Question Find the principal value of each of the following: Answer Let cos y = We need to find the value of y. We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p]. cos y = – cos cos y = y = 1 C. Question Find the principal value of each of the following: Answer Page 2 4. Inverse Trigonometric Functions Exercise 4.1 1 A. Question Find the principal value of each of the following: Answer Let Then sin y = We know that the principal value of is Therefore the principal value of is 1 B. Question Find the principal value of each of the following: Answer Let cos y = We need to find the value of y. We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p]. cos y = – cos cos y = y = 1 C. Question Find the principal value of each of the following: Answer 1 D. Question Find the principal value of each of the following: Answer 1 E. Question Find the principal value of each of the following: Answer Let Then Page 3 4. Inverse Trigonometric Functions Exercise 4.1 1 A. Question Find the principal value of each of the following: Answer Let Then sin y = We know that the principal value of is Therefore the principal value of is 1 B. Question Find the principal value of each of the following: Answer Let cos y = We need to find the value of y. We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p]. cos y = – cos cos y = y = 1 C. Question Find the principal value of each of the following: Answer 1 D. Question Find the principal value of each of the following: Answer 1 E. Question Find the principal value of each of the following: Answer Let Then We know that the principal value of is Therefore the principal value of is . 1 F. Question Find the principal value of each of the following: Answer Let y = Therefore, sin y = We know that the principal value of is And Therefore the principal value of is . 2 A. Question Find the principal value of each of the following: Answer 2 B. Question Find the principal value of each of the following: Answer Page 4 4. Inverse Trigonometric Functions Exercise 4.1 1 A. Question Find the principal value of each of the following: Answer Let Then sin y = We know that the principal value of is Therefore the principal value of is 1 B. Question Find the principal value of each of the following: Answer Let cos y = We need to find the value of y. We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p]. cos y = – cos cos y = y = 1 C. Question Find the principal value of each of the following: Answer 1 D. Question Find the principal value of each of the following: Answer 1 E. Question Find the principal value of each of the following: Answer Let Then We know that the principal value of is Therefore the principal value of is . 1 F. Question Find the principal value of each of the following: Answer Let y = Therefore, sin y = We know that the principal value of is And Therefore the principal value of is . 2 A. Question Find the principal value of each of the following: Answer 2 B. Question Find the principal value of each of the following: Answer 3 A. Question Find the domain of each of the following functions: f(x) = sin –1 x 2 Answer Domain of lies in the interval [–1, 1]. Therefore domain of lies in the interval [–1, 1]. –1 1 But cannot take negative values, So, 0 1 –1 1 Hence domain of is [–1, 1]. 3 B. Question Find the domain of each of the following functions: f(x) = sin –1 x + sinx Answer Domain of lies in the interval [–1, 1]. –1 1. The domain of sin x lies in the interval –1.57 1.57 From the above we can see that the domain of sin –1 x + sinx is the intersection of the domains of sin –1 x and sin x. So domain of sin –1 x + sinx is [–1, 1]. 3 C. Question Find the domain of each of the following functions: f(x) = Answer Domain of lies in the interval [–1, 1]. Therefore, Domain of lies in the interval [–1, 1]. –1 1 Page 5 4. Inverse Trigonometric Functions Exercise 4.1 1 A. Question Find the principal value of each of the following: Answer Let Then sin y = We know that the principal value of is Therefore the principal value of is 1 B. Question Find the principal value of each of the following: Answer Let cos y = We need to find the value of y. We know that the value of cos is negative for the second quadrant and hence the value lies in [0, p]. cos y = – cos cos y = y = 1 C. Question Find the principal value of each of the following: Answer 1 D. Question Find the principal value of each of the following: Answer 1 E. Question Find the principal value of each of the following: Answer Let Then We know that the principal value of is Therefore the principal value of is . 1 F. Question Find the principal value of each of the following: Answer Let y = Therefore, sin y = We know that the principal value of is And Therefore the principal value of is . 2 A. Question Find the principal value of each of the following: Answer 2 B. Question Find the principal value of each of the following: Answer 3 A. Question Find the domain of each of the following functions: f(x) = sin –1 x 2 Answer Domain of lies in the interval [–1, 1]. Therefore domain of lies in the interval [–1, 1]. –1 1 But cannot take negative values, So, 0 1 –1 1 Hence domain of is [–1, 1]. 3 B. Question Find the domain of each of the following functions: f(x) = sin –1 x + sinx Answer Domain of lies in the interval [–1, 1]. –1 1. The domain of sin x lies in the interval –1.57 1.57 From the above we can see that the domain of sin –1 x + sinx is the intersection of the domains of sin –1 x and sin x. So domain of sin –1 x + sinx is [–1, 1]. 3 C. Question Find the domain of each of the following functions: f(x) = Answer Domain of lies in the interval [–1, 1]. Therefore, Domain of lies in the interval [–1, 1]. –1 1 0 1 1 2 and Domain of is [– [1, . 3 D. Question Find the domain of each of the following functions: f(x) = sin –1 x + sin –1 2x Answer Domain of lies in the interval [–1, 1]. –1 1 Therefore, the domain of lies in the interval –1 1 The domain of sin –1 x + sin –1 2x is the intersection of the domains of sin –1 x and sin –1 2x. So, Domain of sin –1 x + sin –1 2x is . 4. Question If sin –1 x + sin –1 y + sin –1 z + sin –1 t = 2p, then find the value of x 2 + y 2 + z 2 + t 2 . Answer Range of sin –1 x is . Give that sin –1 x + sin –1 y + sin –1 z + sin –1 t = 2p Each of sin –1 x, sin –1 y, sin –1 z, sin –1 t takes value of . So, x = 1, y = 1, z = 1 and t = 1. Hence, = x 2 + y 2 + z 2 + t 2 = 1 + 1 + 1 + 1 = 4 5. Question If (sin –1 x) 2 + (sin –1 y) 2 + (sin –1 z) 2 = 3/4 p 2 . Find x 2 + y 2 + z 2 . AnswerRead More
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