Ncert Solutions for Class 12 Inverse Trigonometric Functions

Ncert Solutions for Class 12

Inverse Trigonometric Functions


Inverse Trigonometric Functions

Exercise 2.1

Find the principal values of the following:

1. (sin^{-1}({-frac{1}{2}}))

2. (cos^{-1}({frac{sqrt{3}}{2}}))

3. (cosec^{-1}(2))

4. (tan^{-1}(-sqrt{3}))

5. (cos^{-1}({-frac{1}{2}}))

6. (tan^{-1}(-1))

7. (sec^{-1}({frac{2}{sqrt 3}}))

8. (cot^{-1}(sqrt 3))

9. (cos^{-1}({-frac{1}{sqrt 2}}))

10. (cosec^{-1}(-sqrt{2}))

Find the values of the following:

11. (tan^{-1}(1))+(cos^{-1}({-frac{1}{sqrt 2}}))+(sin^{-1}({-frac{1}{sqrt 2}})).

12. (cos^{-1}({frac{1}{sqrt 2}}))+2(sin^{-1}({frac{1}{sqrt 2}})).

13. If (sin^{-1}x=y), then

(A) (0le ylepi)

(B) (-frac{pi}{2}le y le -frac{pi}{2})

(C) (0 < y <pi)

(D) (-frac{pi}{2} < y < -frac{pi}{2}) 

14. (tan^{-1}{sqrt{3}}-sec^{-1}(-2)) is equal to:

(A) (pi)

(B) (-frac{pi}{3})

(C) (frac{pi}{3})

(D) (frac{2pi}{3})

Inverse Trigonometric Functions

Exercise 2.2

Prove the following:

1. (3sin^{-1}{x}=sin^{-1}(3x-4x^3)), (xin[-frac{1}{2},frac{1}{2}]).

2. (3cos^{-1}{x}=cos^{-1}(4x^3-3x)), (xin[frac{1}{2},1]).

Write the following functions in the simplest form:

3. (tan^{-1}{frac{sqrt{1+x^2}-1}{x}}), (xne {0}).

4. (tan^{-1}(sqrt{frac{1-cosx}{1+cosx}})),(0<x<pi).

5. (tan^{-1}(frac{cosx-sinx}{cosx+sinx})), (-frac{pi}{4}<x<frac{3pi}{4}).

6. (tan^{-1}{frac{x}{sqrt{a^2-x^2}}}), (|x|<a).

7. (tan^{-1}(frac{3a^2x-x^3}{a^3-3ax^2})), (a>0;-frac{a}{sqrt3}<x<frac{a}{sqrt3}).

Find the values of each of the following:

8. (tan^{-1}[2cos(2sin^{-1}{frac{1}{2}})]).

9. (tan^{-1}[sin^{-1}{frac{2x}{1+x^2}}+cos^{-1}{frac{1-y^2}{1+y^2}}]), (|x|<1, y>0) and (xy<1).

Find the values of each of the expressions in Exercises 10 to 15.

10. (sin^{-1}(sin{frac{2pi}{3}})).

11. (tan^{-1}(tan{frac{3pi}{4}})).

12. (tan(sin^{-1}{frac{3}{5}}+cot^{-1}{frac{3}{2}})).

13. (cos^{-1}(cos{frac{7pi}{6}})).

(A) (frac{7pi}{6})

(B) (frac{5pi}{6})

(C) (frac{pi}{3})

(D) (frac{pi}{6})

14. (sin(frac{pi}{3}-sin^{-1}(-frac{1}{2}))).

(A) (frac{1}{2})

(B) (frac{1}{3})

(C) (frac{1}{4})

(D) (1)

15. (tan^{-1}{sqrt3}-cot^{-1}({-sqrt3})) is equal to:

(A) (pi)

(B) (-frac{pi}{2})

(C) (0)

(D) (2sqrt3)


Inverse Trigonometric Functions

Miscellaneous Exercise

Find the value of the following:

1. (cos^{-1}(cosfrac{13pi}{6})).

2. (tan^{-1}(tanfrac{7pi}{6})).

3.  (2sin^{-1}{frac{3}{5}})(=tan^{-1}{frac{24}{7}}).

4. (sin^{-1}{frac{8}{17}}+sin^{-1}{frac{3}{5}})(=tan^{-1}{frac{77}{36}}).

5. (cos^{-1}{frac{4}{5}}+cos^{-1}{frac{12}{13}})(=cos^{-1}{frac{33}{65}}).

6. (cos^{-1}{frac{12}{13}}+sin^{-1}{frac{3}{5}})(=sin^{-1}{frac{56}{65}}).

7. (tan^{-1}{frac{63}{16}}=sin^{-1}{frac{5}{13}})(+cos^{-1}{frac{3}{5}}).

Prove that:

8. (tan^{-1}{sqrt{x}}=frac{1}{2}cos^{-1}{frac{1-x}{1+x}}).

9. (cot^{-1}(frac{sqrt{1+sinx}+sqrt{1-sinx}}{sqrt{1+sinx}+sqrt{1-sinx}}))(=frac{x}{2},;xin(0,frac{pi}{4})).

10. (tan^{-1}(frac{sqrt{1+x}+sqrt{1-x}}{sqrt{1+x}+sqrt{1-x}}))(=frac{pi}{4}-frac{1}{2}cos^{-1}x),(;-frac{1}{sqrt2}le{x}le{1}).

Solve the following equations:

11. (2tan^{-1}(cosx)=tan^{-1}(2cosecx)).

12. (tan^{-1}{frac{1-x}{1+x}})(=frac{1}{2}tan^{-1}x,;(x>0))

13. (sin(tan^{-1}x),|x|<1) is equal to:

(A) (frac{x}{sqrt{1-x^2}})

(B) (frac{1}{sqrt{1-x^2}})

(C) (frac{1}{sqrt{1+x^2}})

(D) (frac{x}{sqrt{1+x^2}})

14. (sin^{-1}(1-x)-2sin^{-1}x)(=frac{pi}{2}), then x is equal to:

(A) (0,frac{1}{2})

(B) (1,frac{1}{2})

(C) (0)

(D) (frac{1}{2})


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