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Inverse Trigonometric Functions Test - 4

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Inverse Trigonometric Functions Test - 4
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  • Question 1
    1 / -0
    If sin-1 for 0 < |x| < , then x is equal to
  • Question 2
    1 / -0
    If sin-1+ cosec-1 = , then the value of x is
  • Question 3
    1 / -0

    The value of \(\cos ^{-1}\left(\cos \frac{5 \pi}{3}\right)+\sin ^{-1}\left(\sin \frac{5 \pi}{3}\right)\) is:

    Solution

    Given:

    \(\cos ^{-1}\left(\cos \left(\frac{5 \pi}{3}\right)\right)+\sin ^{-1}\left(\sin \left(\frac{5 \pi}{3}\right)\right)\)

    We know, 

    \(\cos x=\left(\sin \frac{\pi}{2}-x\right)\)

    So,

    \(=\cos ^{-1}\left(\sin \left(\frac{\pi}{2}-\frac{5 \pi}{3}\right)\right)+\sin ^{-1}\left(\sin \left(\frac{5 \pi}{3}\right)\right) \)

    \(=\cos ^{-1}\left(\sin \left(-\frac{7 \pi}{6}\right)\right)+\sin ^{-1}\left(\sin \left(\frac{5 \pi}{3}\right)\right) \)

    \(=\cos ^{-1}\left(-\sin \left(\frac{7 \pi}{6}\right)\right)+\sin ^{-1}\left(\sin \left(\frac{5 \pi}{3}\right)\right)\)

    Now, 

    \(\sin ^{-1}\left(\sin \frac{5 \pi}{3}\right)=-\frac{\pi}{3}\)....(i)

    Now, 

    \(\cos ^{-1}\left(-\sin \frac{7 \pi}{6}\right)=\frac{\pi}{3}\)....(ii)

    From (i) and (ii),

    \(=\left(\frac{\pi}{3}\right)+\left(-\frac{\pi}{3}\right) \)

    \(=0\)

    Hence, the correct option is (A).

  • Question 4
    1 / -0
    If we consider only the principal values of the inverse trigonometric functions, then the value of tan
  • Question 5
    1 / -0
    If cos (2 sin-1x) =, then x is equal to
  • Question 6
    1 / -0
    The number of real solutions of tan-1 is
  • Question 7
    1 / -0

    What is the value of \(\sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3} ?\)

    Solution

    We know that:

    \(\sin ^{2} x+\cos ^{2} x=1 \)

    \(\tan ^{-1}(\tan x)=x \)

    So,

    Let, \(\sin ^{-1} \frac{4}{5}={x}\)

    \(\Rightarrow \sin x=\frac{4} { 5}\)

    Now, \(\cos ^{2} x=1-\sin ^{2} {x}=1-\frac{16}{25}=\frac{9}{25}\)

    \(\Rightarrow \cos x=\frac{3 }{ 5}\)

    Now, 

    \(\cot x=\frac {\cos x }{ \sin x}=\frac 3  4 \quad \ldots .(1)\) 

    And,

    \(2 \tan ^{-1} \frac{1}{3}=\tan ^{-1}\left(\frac{\frac{2}{3}}{1-\frac{1}{9}}\right) \quad \left(\because 2 \tan ^{-1} {x}=\tan ^{-1}\left(\frac{2 {x}}{1-{x}^{2}}\right)\right)\) 

    \(=\tan ^{-1}\left(\frac{\frac{2}{3}}{\frac{8}{9}}\right)\) 

    \(=\tan ^{-1}\left(\frac{3}{4}\right)\)

     \(=\tan ^{-1}(\cot {x}) \quad  from \ 1\) 

    \(=\tan ^{-1}\left(\tan \left(\frac{\pi}{2}-{x}\right)\right)\)

    \(=\frac{\pi }{ 2}-x \quad \left(\because \tan ^{-1}(\tan x)=x\right)\)

    \(=\frac{\pi}{2}-\sin ^{-1} \frac{4}{5}\)

    \(\therefore \sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3}=\sin ^{-1} \frac{4}{5}+\frac{\pi}{2}-\sin ^{-1} \frac{4}{5}\)

    \(=\frac{\pi}{2}\)

    Hence, the correct option is (B).

  • Question 8
    1 / -0
    cos-1 is equal to
  • Question 9
    1 / -0
    If tan A =  and tan B = , then the value of A + B, i.e. tan–1 + tan–1 is
  • Question 10
    1 / -0
    If cos-1 x + cos-1 y = 2π, then the value of sin-1 x + sin-1 y is
  • Question 11
    1 / -0
    is equal to
  • Question 12
    1 / -0
    is equal to
  • Question 13
    1 / -0
    If , then x is equal to
  • Question 14
    1 / -0
    If cos-1 (1/x) =, then tanwill be
  • Question 15
    1 / -0
    The solution set of the equation sin-1x = 2 tan-1x is
  • Question 16
    1 / -0
    If |a| < 1, |b| < 1 and sin-1  + sin-1  = 2 tan-1 x, then x =
  • Question 17
    1 / -0
    If θ = cos-1 , then tan θ is equal to
  • Question 18
    1 / -0
    For which value of x is sin (cos–1 (x + 1)) = cos (tan–1x)?
  • Question 19
    1 / -0
    tan (cos-1 x) is equal to
  • Question 20
    1 / -0
    If tan-1 a + tan-1b = sin-1 1 - tan-1 c, then
  • Question 21
    1 / -0
    If cos (cos-1 x + sin-1 ) = 0, then x is equal to
  • Question 22
    1 / -0
    The value of sec is
  • Question 23
    1 / -0
    If, then  is equal to
  • Question 24
    1 / -0
    If sin (sin-1 + cos-1 x) = 1, then the value of x is
  • Question 25
    1 / -0
    If 2 tan-1 (cos x) = tan-1 (2 cosec x), then x is equal to
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