Class12 ISC Chapter ITF Exercise 1.1

Class XII ISC Chapter ITF Exercise 1.1

Inverse trigonometric functions are a key topic in class 12 ISC Maths, providing the foundation for solving complex equations involving angles and trigonometric ratios. On this page, you'll find a comprehensive collection of practice questions covering various aspects of inverse trigonometric functions, including basic concepts, properties, and applications. These questions are designed to help students reinforce their understanding and build confidence in tackling problems related to sin^-1(𝑥),cos^-1(𝑥),tan^-1(𝑥) and more.
Each question is carefully crafted to align with the ISC syllabus, ensuring that you're well-prepared for your exams.

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ITF

Class XII ISC Chapter ITF Exercise 1.1

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Give the principal values for the following (1 to 6)

(i) sin-1( 1 / √2 )

Q1(ii) cos-1( 1 / √2 )

Q1(iii) tan-1(√3)

Q2(i) sec-1( 2 / √3 )

Q2(ii) cot-1(√3)

Q2(iii) cosec-1 (√2)

Q3(i) cos-1(- 1 / 2 )

Q3(ii) tan-1(-1)

Q3(iii) cosec-1(√2)

Q4(i) cot-1(- 1 / √3 )

Q4(ii) sec-1 ( 2 / √3 )

Q4(iii) sin-1(-1)

Q5(i) cos-1(cos 4π / 3 )

Q5(ii) sin-1(sin 4π / 3 )

Q5(iii) tan-1(tan 3π / 4 )

Q6(i) cos-1(cos 5π / 3 )

Q6(ii) tan-1(tan 9π / 8 )

Q6(iii) sec-1(sec 9π / 5 )

Show that :

Q7(i) tan-1(tan 5π / 6 ) ≠ 5π / 6 , what is its value ?

Q7(ii) cos-1(cos - π / 6 ) ≠- π / 6 , what is its value ?

Q7(iii) sin-1(sin 5π / 3 ) ≠ 5π / 3 , what is its value ?

Find the domain of the following functions

Q8(i) sin-1(1-x)

Q8(ii) sec-1(2x-3)

Q9 Write the range of one branch of sin-1x , Other than the principal branch.

Q10 Write the range of one branch of cos-1x , Other than the principal branch.

Q11 If cot-1( 3 / 4 ) = x , find the value of cosx

Q12(i) If tan-1( 3 / 4 ) = x , find the value of cosx and sinx

Q12(ii) If cot-1(- 1 / 7 ) = x , find the value of cos x and sin x

Q12(iii) If tan-1x = sin-1 ( 1 / 2 ), find the value of x

Evaluate the following :

Q13(i) cot(tan-1√3)

Q13(ii) sin ( π / 6 - sin-1 (- √3 / 2 ))

Q13(iii) cos (cos-1 (- √3 / 2 ) + π / 6 )

Prove the following :

Q14(i) sin-1 (- 1 / 2 ) + cos-1 (- √3 / 2 ) = 2π / 3

Q14(ii) tan-1(-1) + cos-1(- 1 / √2 ) = π / 3

Using the principal values, find the values of :

Q15(i) cos-1 ( 1 / 2 ) - 2 sin-1 (- 1 / 2 )

Q15(ii) tan-1(√3) - cot-1(- √3)

Q15(iii) tan-1(-1) + cos-1(- 1 / √2 )

Q15(iv) cosec-1(-1) + cot-1(- 1 / √3 )

Find the values of the following :

Q16(i) tan-1(2 cos(2 sin-1 1 / 2 ))

Q16(ii) tan-1(2 sin(2 cos-1 √3 / 2 ))

Find the values of :

Q17(i) tan-1(tan 5π / 6 ) + cos-1(cos 13π / 6 )

Q17(ii) tan-1(- 1 / √3 ) + cot-1( 1 / √3 ) + tan-1(sin (- π / 2 ))

Evaluate the following :

Q18(i) cos(sin-1(- 3 / 5 ))

Q18(ii) cosec(cos-1(- 12 / 13 ))

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(i) sin^-1(
1 / √2
)

Q1(ii) cos^-1(
1 / √2
)

Q1(iii) tan^-1(√3)

Q2(i) sec^-1(
2 / √3
)

Q2(ii) cot^-1(√3)

Q2(iii) cosec^-1 (√2)

Q3(i) cos^-1(-
1 / 2
)

Q3(ii) tan^-1(-1)

Q3(iii) cosec^-1(√2)

Q4(i) cot^-1(-
1 / √3
)

Q4(ii) sec^-1 (
2 / √3
)

Q4(iii) sin^-1(-1)

Q5(i) cos^-1(cos
4π / 3
)

Q5(ii) sin^-1(sin
4π / 3
)

Q5(iii) tan^-1(tan
3π / 4
)

Q6(i) cos^-1(cos
5π / 3
)

Q6(ii) tan^-1(tan
9π / 8
)

Q6(iii) sec^-1(sec
9π / 5
)

Q7(i) tan^-1(tan
5π / 6
) ≠
5π / 6
, what is its value ?

Q7(ii) cos^-1(cos -
π / 6
) ≠-
π / 6
, what is its value ?

Q7(iii) sin^-1(sin
5π / 3
) ≠
5π / 3
, what is its value ?

Q8(i) sin^-1(1-x)

Q8(ii) sec^-1(2x-3)

Q9 Write the range of one branch of sin^-1x , Other than the principal branch.

Q10 Write the range of one branch of cos^-1x , Other than the principal branch.

Q11 If cot^-1(
3 / 4
) = x , find the value of cosx

Q12(i) If tan^-1(
3 / 4
) = x , find the value of cosx and sinx

Q12(ii) If cot^-1(-
1 / 7
) = x , find the value of cos x and sin x

Q12(iii) If tan^-1x = sin^-1 (
1 / 2
), find the value of x

Q13(i) cot(tan^-1√3)

Q13(ii) sin (
π / 6
- sin^-1 (-
√3 / 2
))

Q13(iii) cos (cos^-1 (-
√3 / 2
) +
π / 6
)

Q14(i) sin^-1 (-
1 / 2
) + cos^-1 (-
√3 / 2
) =
2π / 3

Q14(ii) tan^-1(-1) + cos^-1(-
1 / √2
) =
π / 3

Q15(i) cos^-1 (
1 / 2
) - 2 sin^-1 (-
1 / 2
)

Q15(ii) tan^-1(√3) - cot^-1(- √3)

Q15(iii) tan^-1(-1) + cos^-1(-
1 / √2
)

Q15(iv) cosec^-1(-1) + cot^-1(-
1 / √3
)

Q16(i) tan^-1(2 cos(2 sin^-1
1 / 2
))

Q16(ii) tan^-1(2 sin(2 cos^-1
√3 / 2
))

Q17(i) tan^-1(tan
5π / 6
) + cos^-1(cos
13π / 6
)

Q17(ii) tan^-1(-
1 / √3
) + cot^-1(
1 / √3
) + tan^-1(sin (-
π / 2
))

Q18(i) cos(sin^-1(-
3 / 5
))

Q18(ii) cosec(cos^-1(-
12 / 13
))