Consider the following statement -i sin-1 sin2-3 -2-3ii tan-1 tan2-3 -3iii cos-1 cos2-3 -2-3iv sec-1 sec5-4 -3-4

Sin^ 1 ( sin2π/3 )=sin^ 1 ( sin ( π π/3 ) ) =sin^ 1 ( sinπ/3 )=π/3 ∴ (i) is wrong. tan^ 1 (sin 2π/3 )=tan^ 1tan ( π+2π/3 ) =tan^ 1tan ( π/3 )= π/3 ∴ (ii) i ...

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Byju's Answer

Standard XIII

Mathematics

Basic Inverse Trigonometric Functions

Consider the ...

Question

Consider the following statement :
(i) $s i n^{- 1} (s i n \frac{2 π}{3}) = \frac{2 π}{3}$
(ii) $t a n^{- 1} (t a n \frac{2 π}{3}) = \frac{- π}{3}$
(iii) $c o s^{- 1} (c o s \frac{2 π}{3}) = \frac{2 π}{3}$
(iv) $s e c^{- 1} (s e c \frac{5 π}{4}) = \frac{3 π}{4}$

(i), (ii), (iii)

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(ii), (iii), (iv)

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(i), (ii), (iv)

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None of these

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Solution

The correct option is B (ii), (iii), (iv)
$s i n^{- 1} (s i n \frac{2 π}{3}) = s i n^{- 1} (s i n (\frac{π - π}{3}))$
$= s i n^{- 1} (s i n \frac{π}{3}) = \frac{π}{3}$
$∴$ (i) is wrong.
$t a n^{- 1} (s i n \frac{2 π}{3}) = t a n^{- 1} t a n (- π + \frac{2 π}{3})$
$= t a n^{- 1} t a n (\frac{- π}{3}) = \frac{- π}{3}$
$∴$ (ii) is true.
$c o s^{- 1} (c o s \frac{2 π}{3}) = \frac{2 π}{3} a s \frac{2 π}{3} ϵ [0, 2 π]$
$∴$ (iii) is true
$s e c^{- 1} (s e c \frac{5 π}{4}) = s e c^{- 1} s e c (2 π - \frac{3 π}{4})$
$= s e c^{- 1} s e c \frac{3 π}{4} = \frac{3 π}{4}$
$∴$ (iv) is true
Hence (ii), (iii) and (iv) are true.

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Similar questions

Q. Consider the following statement :
(i)

s i n^{- 1} (s i n \frac{2 π}{3}) = \frac{2 π}{3}

(ii)

t a n^{- 1} (t a n \frac{2 π}{3}) = \frac{- π}{3}

(iii)

c o s^{- 1} (c o s \frac{2 π}{3}) = \frac{2 π}{3}

(iv)

s e c^{- 1} (s e c \frac{5 π}{4}) = \frac{3 π}{4}

Q. Using principal values, evaluate the following

{cos}^{- 1} (cos \frac{2 π}{3})

{sin}^{- 1} (sin \frac{2 π}{3})

Q. Find the value of

{cos}^{- 1} (cos \frac{2 π}{3}) + {sin}^{- 1} (sin \frac{2 π}{3})

Q. The principle value of

{cos}^{- 1} (cos \frac{2 π}{3}) + {sin}^{- 1} (sin \frac{2 π}{3})

is:

Q. What is the principal value of

{cos}^{- 1} (cos \frac{2 π}{3}) + {sin}^{- 1} (sin \frac{2 π}{3})

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Consider the following statement : (i) sin−1(sin2π3)=2π3 (ii) tan−1(tan2π3)=−π3 (iii) cos−1(cos2π3)=2π3 (iv) sec−1(sec5π4)=3π4

Consider the following statement :
(i) $s i n^{- 1} (s i n \frac{2 π}{3}) = \frac{2 π}{3}$
(ii) $t a n^{- 1} (t a n \frac{2 π}{3}) = \frac{- π}{3}$
(iii) $c o s^{- 1} (c o s \frac{2 π}{3}) = \frac{2 π}{3}$
(iv) $s e c^{- 1} (s e c \frac{5 π}{4}) = \frac{3 π}{4}$