第12类RD Sharma解-第4章逆三角函数–练习4.1

问题1.找出以下各项的本金值：

(i)罪^-1 (-√3/ 2)

(ii)罪^-1 (cos2π/ 3)

(iii)罪^-1 (-√3– 1 /2√2)

(iv)罪^-1 (√3+ 1 /2√2)

(v)罪^-1 (cos3π/ 4)

(vi)罪^-1 (tan5π/ 4)

解决方案：

   (i) Sin^-1 (-√3/ 2)

  = Sin^-1 [sin (- π / 3)] 

  = – π / 3     Ans. 

  (ii) Sin^-1 (cos 2π / 3)

   = Sin^-1 (- 1 / 2)

   = Sin^-1 (- π / 6)

   = – π / 6   Ans.

 (iii) Sin^-1 (√3 – 1/2√2)

   = Sin^-1 ( sin π / 12 ) 

   = π / 12  Ans. 

 (iv) Sin^-1 (√3 + 1/2 √2) 

   = Sin^-1 (5π / 12)

   = 5π / 12   Ans.

(v) Sin^-1 (cos 3π / 4)

   = Sin^-1 (- √2 / 2)

   = [Sin^-1 (- π / 4)]

   = – π / 4   Ans.

(vi) Sin^-1 (tan 5π / 4)

  = Sin^-1 (1)

  = Sin^-1 [sin (π / 2)]

  = π / 2    Ans.

问题2。

(i)罪^-1 1/2 -2罪^-1 1 /√2

(ⅱ)仙^-1 {COS(茜^-1√3/ 2)}

解决方案：

(i) Sin^-1 1/2  -2  Sin^-1 1/ √2

    = Sin^-1 1/2 – Sin^-1 [ 2 x 1/ √2 √1- $(1 /√2)2$ ] 

    =   Sin^-1 1/2 – Sin^-1 (1)

    =  π/6 –  π /2

    =  π / 3   Ans. 

(ii) Sin^-1 { cos ( Sin^-1 sin  π / 3 )}

   =   Sin^-1 { cos ( π / 3 ) } 

   =    Sin^-1 { 1/2 }

   =   Sin^-1 { sin  π / 6 }

   =  π / 6   Ans.

问题3.找到以下每个功能的领域：

(i)f(x)= Sin ^-1 x2

(ii)f(x)= Sin ^-1 x + sinx

(ⅲ)F(X)= SIN ^-1√x2 – 1

(iv)f(x)= Sin ^-1 x + Sin ^-1 2x

解决方案：

(i) Domain of  Sin-1 lies between the interval [ -1 , 1 ]

and x² ∈ [ 0 , 1 ] as x² can not be negative .

 So, x ∈ [ -1 , 1 ] 

Hence, the domain of the function f(x) = [ -1 , 1 ]   Ans. 

(ii) Let f(x) = g(x) + h(x) , where g(x) =  Sin^-1x  and h(x) = sinx respectively.

Therefore , the domain of f(x) is given by the intersection of the domain g(x) & h(x) .

The domain of g(x) = [ -1 , 1 ]

The domain of h(x) = [ – ∞ , ∞ ] 

Thus, the interaction of g(x) and h(x) is [ -1 , 1 ] 

Hence , the domain of f(x) is [ -1 , 1 ]    Ans.

(iii) As we know , the domain of Sin^-1 x is [ -1 , 1 ]

Therefore , domain of Sin^-1  √x² – 1 will also lies in the interval [ -1 , 1 ]

  :. x² – 1 ∈ [ 0 , 1 ] as square root cannot be negative .

 => x² ∈ [ 1 , 2 ] 

 => x ∈ [ – √2 , -1 ] U [ 1 , √2 ] 

Hence, the domain of function f(x) = [ – √2 , -1 ] U [ 1 , √2 ]   Ans. 

(iv) Let f(x) = g(x) + h(x), where g(x) = Sin^-1 x  x and h(x) = Sin^-1 2x

Therefore, the domain of f(x) will be given by the intersection of g(x) and h(x) .

the domain of g(x) = [ -1 , 1 ] 

lly , the domain of h(x) = [ -1/2 , 1/2 ] 

g(x) ∩ h(x) = [ -1 , 1 ]  ∩ [ -1/2 , 1/2 ]

Hence, the domain of the function f(x) =  [- 1/ 2 , 1/2 ]    Ans. 

问题4.如果sin ^-1 x + sin ^-1 y + sin ^-1 z + sin ^-1 t =2π，则求出x ² + y ² + z ² + t ^2的值。

解决方案：

As we already know, Range of sin^-1 is [ – π / 2 , π / 2 ] 

 Given: (sin^-1x) + (sin^-1y) +(sin^-1y)+(sin^-1t) = 2 π

So, each takes the value of π / 2 

:. x = 1, y = 1, z = 1 & t = 1 

Hence, x² + y²+ z² + t² = 1+ 1 + 1 + 1 = 4    Ans. 

问题5.如果(sin ^-1 x) ² +(sin ^-1 y) ² +(sin ^-1 y) ² =3π2/ 4，则求出x ² + y ² + z ^2的值。

解决方案：

As we already know , Range of   sin-1 is [ – π / 2 , π / 2 ]

Given:     ( sin^-1x )² + ( sin^-1y )² + ( sin^-1y )² = 3π2/4 

:. each takes the value of  π / 2

x = 1, y = 1 & z = 1 .

Hence ,  x² + y² + z² = 1 + 1 + 1 = 3    Ans.