Class 11th Maths – RD Sharma Chapter 5 : trigonometric Functions Exercise 5.3 Solutions

1. 1. Find the values of the following trigonometric ratios:

   (i) sin 5π/3 Watch Solution

   (ii) sin 17π Watch Solution

   (iii) tan 11π/6 Watch Solution

   (iv) cos (−25π/4) Watch Solution

   (v) tan 7π/4 Watch Solution

   (vi) sin 17π/6 Watch Solution

   (vii) cos 19π/6 Watch Solution

   (viii) sin (−11π/6) Watch Solution

   (ix) cosec (−20π/3) Watch Solution

   (x) tan (−13π/4) Watch Solution

   (xi) cos 19π/4 Watch Solution

   (xii) sin 41π/4 Watch Solution

   (xiii) cos 39π/4  Watch Solution

   (xiv) sin 151π/6 Watch Solution

   2. Prove that:

   (i) tan 225° cot 405° + tan 765° cot 675° = 0 Watch Solution

   (ii) sin 8π/3 cos 23π/6 + cos 13π/3 sin 35π/6 = 1/2 Watch Solution

   (iii) cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = 1/2 Watch Solution

   (iv) tan (−225°) cot (−405°) − tan (−765°) cot (675°) = 0 Watch Solution

   (v) cos 570° sin 510° + sin (−330°) cos (−390°) = 0  Watch Solution

   (vi) tan 11π/3 − 2 sin 4π/6 − (3/4) cosec^2 π/4 + 4 cos^2 17π/6 = (3 − 4√3)/2 Watch Solution

   (vii) 3 sin π/6 sec π/3 − 4 sin 5π/6 cot π/4 = 1 Watch Solution

   3. Prove that:

   (i) [cos (2π + x) cosec (2π + x) tan (π/2 + x)] / [sec (π/2 + x) cos x cot (π + x)] = 1 Watch Solution

   (ii) [cosec (90° + x) + cot (450° + x)] / [cosec (90° − x) + tan (180° − x)] + [tan (180° + x) + sec (180° − x)] / [tan (360° + x) − sec (−x)] = 2 Watch Solution

   (iii) [sin (π + x) cos (π/2 + x) tan (3π/2 − x) cot (2π − x)] / [sin (2π − x) cos (2π + x) cosec (−x) sin (3π/2 − x)] = 1 Watch Solution

   (iv) {1 + cot x − sec (π/2 + x)} {1 + cot x + sec (π/2 + x)} = 2 cot x  Watch Solution

   (v) [tan (π/2 − x) sec (π − x) sin (−x)] / [sin (π + x) cot (2π − x) cosec (π/2 − x)] = 1 Watch Solution

   4. Prove that:

   sin^2 (π/18) + sin^2 (π/9) + sin^2 (7π/18) + sin^2 (4π/9) = 2 Watch Solution

   5. Prove that:

   sec (3π/2 − x) sec (x − 5π/2) + tan (5π/2 + x) tan (x − 3π/2) = −1 Watch Solution

   6. In a ΔABC, prove that:

   (i) cos (A + B) + cos C = 0 Watch Solution

   (ii) cos [(A + B)/2] = sin (C/2) Watch Solution

   (iii) tan [(A + B)/2] = cot (C/2) Watch Solution

   7. If A, B, C, D be the angles of a cyclic quadrilateral, taken in order, prove that:

   cos (180° − A) + cos (180° + B) + cos (180° + C) − sin (90° + D) = 0 Watch Solution

   8. Find x from the following equations:

   (i) cosec (π/2 + θ) + x cos θ cot (π/2 + θ) = sin (π/2 + θ) Watch Solution

   (ii) x cot (π/2 + θ) + tan (π/2 + θ) sin θ + cosec (π/2 + θ) = 0 Watch Solution

   9. Prove that:

   (i) tan 4π − cos 3π/2 − sin 5π/6 cos 2π/3 = 1/4 Watch Solution

   (ii) sin 13π/3 sin 8π/3 + cos 2π/3 sin 5π/6 = 1/2 Watch Solution

   (iii) sin 13π/3 sin 2π/3 + cos 4π/3 sin 13π/6 = 1/2 Watch Solution

   (iv) sin 10π/3 cos 13π/6 + cos 8π/3 sin 5π/6 = −1 Watch Solution

   (v) tan 5π/4 cot 9π/4 + tan 17π/4 cot 15π/4 = 0 Watch Solution