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Inverse Trigonometric Functions

Short Answer Type

31. Show that cos (sin–¹.x) = sin (cos^–1x) = $square root of 1 minus straight x squared end root space for space open vertical bar straight x close vertical bar space space less or equal than space 1$

Let sin^-1 x= $Let space space space sin to the power of negative 1 end exponent space straight x space equals space straight theta therefore space space space space straight x element of space left square bracket negative 1 comma space 1 right square bracket space and space straight theta element of open square brackets negative straight pi over 2 comma space straight pi over 2 close square brackets therefore space space space straight x equals sin space straight theta space and space straight theta element of space open square brackets negative straight pi over 2 comma space straight pi over 2 close square brackets therefore space space space cos space straight theta space is space plus space ve Now space cos space straight theta space equals space square root of 1 minus sin squared space straight theta end root equals square root of 1 minus straight x squared end root$
[ + ve sign taken before square root]

$therefore space space space cos space left parenthesis sin to the power of negative 1 end exponent straight x right parenthesis equals square root of 1 minus straight x squared end root$

Again, let cos ^–1x = ϕwhere x ∈ [– 1, 1 ] and ϕ ∈ [0,π]
∴ x = cos ϕ where ϕ ∈ [0, π]
∴sin ϕ is + ve

sin space straight ϕ space equals space square root of 1 minus cos squared space straight ϕ end root space equals square root of 1 minus straight x squared end root therefore space space space space sin space left parenthesis cos to the power of negative 1 end exponent straight x right parenthesis equals square root of 1 minus straight x squared end root

(+ ve sign taken before under root)

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32. If sin^–1x = y, then

(A) 0 ≥ y ≥

straight pi

(B)

negative straight pi over 2 less or equal than straight Y less or equal than straight pi over 2

straight pi

(D)

negative straight pi over 2 less than straight y less than straight pi over 2

265 Views

33.

tan to the power of negative 1 end exponent square root of 3 minus s e c to the power of negative 1 end exponent left parenthesis negative 2 right parenthesis

is equal

(A)

straight pi

(B)

negative straight pi over 3

(C)

straight pi over 3

(D)

fraction numerator 2 straight pi over denominator 3 end fraction

130 Views

34. Find the value of sin^–1

open parentheses sin space fraction numerator 3 straight x over denominator 5 end fraction close parentheses.

100 Views

35. Find the values of each of the expressions :

sin to the power of negative 1 end exponent open parentheses sin fraction numerator 2 straight pi over denominator 3 end fraction close parentheses

110 Views

36. Find the values of each of the expressions :

tan to the power of negative 1 end exponent open parentheses tan space fraction numerator 3 straight pi over denominator 4 end fraction close parentheses

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37.

Find the value of the following :

$cos to the power of negative 1 end exponent open parentheses cos space fraction numerator 13 straight pi over denominator 6 end fraction close parentheses$

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38.

Find the value of the following :

$tan to the power of negative 1 end exponent open parentheses tan space fraction numerator 7 space straight pi over denominator 6 end fraction close parentheses$