Q5. cot x = 3/4 and x lies in the third quadrant. Find the other five functions (NCERT Exercise 3.2, Q3).
tan x = 4/3. cosec2 x = 1 + 9/16 = 25/16, and sin x < 0 in the third quadrant, so cosec x = −5/4 and sin x = −4/5. Then cos x = cot x sin x = (3/4)(−4/5) = −3/5 and sec x = −5/3.
Q6. tan x = −5/12 and x lies in the second quadrant. Find the other five functions (NCERT Exercise 3.2, Q5).
cot x = −12/5. sec2 x = 1 + 25/144 = 169/144, and cos x < 0 in the second quadrant, so sec x = −13/12 and cos x = −12/13. Then sin x = tan x cos x = (−5/12)(−12/13) = 5/13 and cosec x = 13/5.
Q7. Find the range of y = 3 + 2 sin x.
−1 ≤ sin x ≤ 1, so 1 ≤ 3 + 2 sin x ≤ 5, and both ends occur (at x = 3π/2 and x = π/2). The range is [1, 5].
Q8. Without a calculator, decide the signs of sin 4 and cos 4 (the angle is 4 radians).
π ≈ 3.14 and 3π/2 ≈ 4.71, so 4 radians lies in the third quadrant: sin 4 < 0 and cos 4 < 0 (so tan 4 > 0).
Q9. Find all real x with (i) sin x = 0; (ii) cos x = 0; (iii) tan x = 0.
(i) x = nπ; (ii) x = (2n + 1)π/2; (iii) tan x = 0 exactly when sin x = 0 (and cos x ≠ 0, which then holds), so x = nπ. Here n is any integer.