More practice questions

5. Q5. Find the principal values of (i) cot^⁻¹(−1); (ii) sec^⁻¹(−√2); (iii) cosec^⁻¹(−2).

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   (i) cot^⁻¹ gives an angle in (0, π): cot(3π/4) = −1, so 3π/4. (ii) cos y = −1/√2 with y in [0, π] − {π/2}: 3π/4. (iii) sin y = −1/2 with y in [−π/2, π/2] − {0}: −π/6.

6. Q6. Find the value of sin^⁻¹(−√3/2) + cos^⁻¹(−√3/2).

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   sin^⁻¹(−√3/2) = −π/3 and cos^⁻¹(−√3/2) = π − π/6 = 5π/6. The sum is −2π/6 + 5π/6 = π/2.

7. Q7. Find the domain of (i) f(x) = sin^⁻¹(2x − 1); (ii) g(x) = cos^⁻¹(x/3).

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   The input of sin^⁻¹ or cos^⁻¹ must lie in [−1, 1]. (i) −1 ≤ 2x − 1 ≤ 1 gives 0 ≤ x ≤ 1: domain [0, 1]. (ii) −1 ≤ x/3 ≤ 1 gives −3 ≤ x ≤ 3: domain [−3, 3].

8. Q8. Find the range of y = 2 sin^⁻¹ x + π/2.

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   −π/2 ≤ sin^⁻¹ x ≤ π/2, so −π ≤ 2 sin^⁻¹ x ≤ π and −π/2 ≤ y ≤ 3π/2. The range is [−π/2, 3π/2].

9. Q9. Which of these exist? (i) cos^⁻¹(π/3); (ii) cosec^⁻¹(−0.6); (iii) tan^⁻¹(100); (iv) cosec^⁻¹(−1).

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   (i) No: π/3 ≈ 1.05 is not in [−1, 1]. (ii) No: cosec^⁻¹ needs |x| ≥ 1. (iii) Yes, tan^⁻¹ is defined on all of R; it is just below π/2. (iv) Yes: cosec^⁻¹(−1) = −π/2.

10. Q10. Give two angles other than the principal value whose sine is −1/2, and say which branch of sin^⁻¹ each belongs to.

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    7π/6 and 11π/6 both have sine −1/2. 7π/6 lies in [π/2, 3π/2], the branch next to the principal one; 11π/6 lies in [3π/2, 5π/2]. The principal value is −π/6, the only one in [−π/2, π/2].