## NCERT Exemplar Problems Class 11 Mathematics Chapter 3  Trigonometric Functions

Q4. If cos (Î± + ) =4/5 and sin (Î±- )=5/13 , where Î± lie between 0 and Ï€/4, then  find the value of tan 2Î±.

Q6. Prove that cos cos /2- cos 3 cos 9/2 = sin 7/2 sin 4 .

Q7. If a cos Î¸ + b sin Î¸ =m and a sin Î¸ -b cosÎ¸ = n, then show that a2 + b2-m2 + n2

Sol: We have, a cos Î¸ + b sin Î¸ = m (i)
and a sin Î¸ -bcos Î¸ = n (ii)

Q8. Find the value of tan 22 °30′

Q9. Prove that sin 4A = 4 sin A cos3A – 4 cos A sin3 A.

Sol: L.H.S. = sin 4A
= 2 sin 2A- cos 2A = 2(2 sin A cosA)(cos2 A – sin2 A)
= 4 sin A • cos3 A – 4 cos A sin3 A = R.H.S.

Q10. If tan + sin = m and tan – sin = n, then prove that m2-n2 = 4 sin tan

Sol:We have, tan + sin = m   (i)
And tan -sin =n  (ii)
Now,                 m + n = 2 tan
And                   m – n = 2 sin.
(m + n)(m -n) = 4 sin 6
tan m2 -n2 = 4 sin -tan

Q11. If tan (A + B) =p and tan (A – B) = q, then show that tan 2A = p+q / 1 – pq

Sol: We have tan (A + B) =p and tan (A – B) = q
tan2A = tan [(A + B) + (A-B)]

Q12. If cos + cos = 0 = sin + sin Î², then prove that cos 2 + cos 2Î² = -2 cos (Î± + ).

Q15.  If sin Î¸+ cos Î¸ =1, then find the general value of Î¸

Q16. Find the most general value of Î¸ satisfying the equation tan Î¸ = -1 and  cos Î¸ = 1/âˆš2 .
Sol:
We have tan Î¸ = -1 and cos Î¸ =1/âˆš2 .
So, Î¸ lies in IV quadrant.
Î¸ = 7/4
So, general solution is Î¸ = 7Ï€/4 + 2 n Ï€, nâˆˆ Z

Q17. If cot Î¸ + tan Î¸ = 2 cosec Î¸, then find the general value of Î¸
Sol:
Given that, cot Î¸ + tan Î¸ = 2 cosec Î¸

Q18. If 2 sin2 Î¸ =3 cos Î¸, where Oâ‰¤Î¸â‰¤2, then find the value of Î¸

Q19. If sec x cos 5x + 1 = 0, where 0 < x <Ï€/2 , then find the value of x.

Q20. If sin(Î¸ + Î±) = a and sin(Î¸ + Î²) = b , then prove that cos2(Î± â€“ Î²) â€“ 4abcos(Î± â€“ Î²) = 1-2a2 -2b2

Sol: We have sin(Î¸ + Î±) = a —(i)
sin(Î¸ + Î²) = b ——-(ii)

Q22. Find the value of the expression

Q23. If a cos 2+b sin 2 = c has Î± and Î² as its roots, then prove that tan Î± +tan Î² = 2b/a+c

Q24. If x = sec Ï•-tanÏ•andy = cosec Ï• + cot Ï• then show that xy + x -y +1=0.

Q25. If lies in the first quadrant and cos =8/17 , then find the value of cos (30 ° + ) + cos (45 ° – ) + cos (120 ° – ).

Q26. Find the value of the expression cos4 Ï€/8 + cos4 3Ï€/8  + cos4 5Ï€/8  + cos47Ï€/8

Q27. Find the general solution of the equation 5 cos2 +7 sin2 -6 = 0.

Q28. Find the general solution ofâ€˜the equation sin x – 3 sin 2x + sin 3x = cos x – 3 cos 2x + cos 3x.
Sol: We have, (sin x + sin 3x) – 3 sin 2x = (cos x + cos 3x) – 3 cos 2x
=> 2 sin 2x cos x – 3 sin 2x = 2 cos 2x.cos x – 3 cos 2x
=> sin 2x(2 cos x – 3) = cos 2x(2 cos x – 3)
=> sin 2x = cos 2x (As cos x â‰  3/2)
=>                           tan 2x = 1       => tan 2x = tan Ï€/4
=>                           2x = nÏ€ + Ï€/4 , nâˆˆZ
x = nÏ€/2 +Ï€/8 , nâˆˆZ

Q29. Find the general solution of the equation (âˆš3- l)cos + (âˆš3+ 1)sin = 2.

Objective Type Questions

Q30. If sin + cosec =2, then sin2 + cosec2 is equal to
(a) 1
(b) 4
(c) 2
(d) None of these

Q31. If f(x) = cos2 x + sec2 x, then â€˜
(a) f(x) <1
(b) f(x) = 1
(c) 2 <f(x) < 1
(d) fx) â‰¥ 2

Q32. If tan Î¸ = 1/2 and tan Ï• = 1/3, then the value of Î¸ + Ï• is

Q33. Which of the following is not correct?

(a) sin Î¸ = – 1/5 (b) cos Î¸ = 1                                 (c) sec Î¸ = -1/2                 (d) tan Î¸ = 20
Sol: (c)
We know that, the range of sec Î¸ is R – (-1, 1).
Hence, sec Î¸ cannot be equal to -1/2

Q34. The value of tan 1 ° tan 2 ° tan 3 ° … tan 89 ° is
(a) 0
(b) 1
(c) 1/2
(d) Not defined

Sol: (b)
tan 1 ° tan 2 ° tan 3 ° … tan 89 °
= [tan 1 ° tan 2 ° … tan 44 °] tan 45 °[tan (90 ° – 44 °) tan (90 ° – 43 °)… tan (90 ° – 1 °)]
= [tan 1 ° tan 2 ° … tan 44 °] [cot 44 ° cot 43 °……. cot 1 °]
= 1-1… 1-1 = 1

Q36. The value of cos 1 ° cos 2 ° cos 3 ° … cos 179 ° is
(a) 1/âˆš2
(b) 0
(c) 1
(d) -1

Sol: (b)
Since cos 90 ° = 0, we have
cos 1 ° cos 2 ° cos 3 ° …cos 90 °… cos 179 ° = 0

Q37. If tan Î¸ = 3 and Î¸ lies in the third quadrant, then the value of sin Î¸ is

Q38. The value of tan 75 ° – cot 75 ° is equal to

Q39. Which of the following is correct?
(a) sin 1 ° > sin 1
(b) sin 1 ° < sin 1
(c) sin l ° = sin l
(d) sin l ° = Ï€/18 ° sin 1

Sol: We know that, in first quadrant if Î¸ is increasing, then sin Î¸ is also increasing.
âˆ´sin 1 ° < sin 1 [âˆµ 1 radian = 57â—¦30′]

Q41. The minimum value of 3 cos x + 4 sin x + 8 is
(a) 5
(b) 9
(c) 7
(d) 3
Sol: (d)
3 cos x + 4sin x + 8 = 5 (3/5 cos x + 4/5sin x) + 8
= 5(sin Î± cos x + cos Î± sin x) + 8
= 5 sin(Î± + x) + 8, where tan Î± = 3/4

Q42. The value of tan 3A – tan 2A – tan A is
(a) tan 3A . tan 2A . tan A
(b) -tan 3A .tan 2A . tan A
(c) tan A . tan 2A – tan 2A . tan 3A – tan 3A . tan A
(d) None of these
Sol:  (a)
3A= A+ 2A
=> tan 3A = tan (A + 2A)
=> tan 3 A = tanA + tan2A/ 1 – tan A . tan 2A
=> tan A + tan 2A = tan 3A – tan 3A• tan 2A . tan A
=> tan 3 A – tan 2A – tan A = tan 3A . tan 2A . tan A

Q43. The value of sin (45 ° + )- cos (45 ° – ) is
(a) 2 cos
(b) 2 sin
(c) 1
(d) 0
Sol: (d)
sin (45 ° + ) – cos (45 ° – ) = sin (45 ° + ) – sin (90 ° – (45 ° – ))
= sin (45 ° + ) – sin (45 °+ ) = 0

Q44. The value of (Ï€/4+ ) cot (Ï€/4- ) is
(a) -1
(b)  0
(c)  1
(d)     Not defined

Q46. The value of cos 12 ° + cos 84 ° + cos 156 ° + cos 132 ° is
(a) 1/2
(b) 1
(c) -1/2
(d) 1/8

Q47. If tan A = 1/2 and tan B = 1/3 then tan (2A + B) is equal to
(a) 1
(b) 2
(c) 3
(d) 4

Q49. The value of sin 50 ° – sin 70 ° + sin 10 ° is equal to
(a) 1
(b) 0
(c) 1
(d) 2

Q50. If sin + cos =1, then the value of sin 2 is
(a) 1
(b) 1
(c) 0
(d) -1