Ch 8 _trigonometry Class X

ASSIGNMENT CLASS X

TRIGONOMETRY

Q1. In  ABC, right angled at B, if AB = 12 cm and BC = 5 cm, find (i) sin A and tan A (ii) sin C and cot C. 20 Q2. Given cot θ  find all other trigonometric ratios. 21 Q3. If cos A 

12 35 verify that: sin A(1  tan A)  . 13 156

Q4. (i) If 7 cot   24, prove that

1  cos  1  1  cos  7

Q6. If 21cosec  = 29, find the value of : Q7. If tan  

5sin   3cos  7  . 5sin   2 cos  2 2 cos2   1 (ii) 2 cos   sin 2 

(ii) If 4 cot   5 , show that: (i)

cos2   sin 2  1  2sin 2 

1 1  2; show that: tan 2   2. tan  tan 2 

Q8. Evaluate each of the following : (i) 2 cos2 60 cot 30° + 6 sin2 30° cosec2 60° (iii) 2 (cos2 45° + tan2 60°) – 6 (sin245° – tan230°) Q9. If  = 30°, verify that :

(i) sin 2 

2 tan  1  tan 2 

5sin 2 30  cos 2 45  4 tan 2 30 2sin 30 cos 30  tan 45 tan 2 60  3sec 2 30  4cos 2 45  5cos 2 90 (iv) cosec 30  sec 60  cot 2 30

(ii)

(ii) cos 2 

1  tan 2  1  tan 2 

(iii) tan 2 

2 tan  1  tan 2 

Q10. Given that sin (A + B) = sin A cos B + cos A sin B, find the value of sin 75°. Q11. If sin (A  2B) 

3 and cos (A + 4B) = 0, find the values of angles A and B. 2

Q12. ABCD is a rectangle with AD =12 cm and DC = 20 cm as shown. The line segment DE is drawn making an angle of 30° with AD, intersecting AB in E. Find the lengths of DE and AE. 12 cm

A

E

B

30°

D

20 cm

C

Q13. Evaluate each of the following: cos 2 20  cos 2 70 sin 2 57  sin 2 33

(i)

2

sin 27   cos 63  (ii)      cos 63   sin 27 

2

(iii) cot 12° cot 38° cot 52° cot 60° cot 78°

Q14. Prove that: (i)

sin .cos (90   cos  cos  sin (90  ).sin   1 sin (90  ) cos (90  )

(ii)

cos (90  ) . sec (90  ) . tan  tan (90  )  2 cos ec (90  ) sin (90  ) .cot (90  ) cot 

Q15. Without using trigonometric tables, find the value of each of the following: (i) (ii)

cos (40  )  sin (50  ) 

sec2 10  cot 2 80 

cos 2 40  cos 2 50 sin 2 40  sin 2 50

sin15 cos 75  cos15 sin 75 cos  sin(90  )  sin  . cos (90  )

(iii)

 tan  cot (90  )  sec  cosec (90  )  sin 2 35  sin 2 55 tan10 tan 20 tan 45 tan 70 tan 80

(iv)

 tan 20   cot 20   cosec70    sec 70   2 tan15 tan 37 tan 53 tan 60 tan 75    

(v)

sec39 2  .tan17 tan 38 tan 60 tan 52 tan 73  3(sin 2 31  sin 2 59) cosec 51 3

2

2

Q16. If sec 5 θ = cosec ( θ – 36°), where 5 θ is an acute angle, find the value of θ . Q17. Simplify the following expressions: (i) (1 + cos ) (cosec  – cot )

(ii) cosec  (1+ cos ) (cosec  – cot )

Q18. Prove that following identities: (i) cosec2 + sec2 = cosec2 . sec2 

iii)

1  tan 2   1  tan     1  cot 2   1  cot  



2

(iv) (vi)

(vii)

sin  sin   2 cot   cosec  cot   c osec 

(viii) 1 

1  1  1  1   tan 2 A   cot 2 A  sin 2 A  sin 4 A

cos2  sin 3    1  sin  cos  1  tan  sin   cos 

sin  cos    1  sin  cos  1  cot  cos   sin 

(x)

(xi)

1  cos   sin 2   cot  sin (1  cos )

(xii)

1 1 1 1    cosec A  cot A sin A sin A c osec A  cot A

cos 2 B  cos 2 A sin 2 A  sin 2 B  (xiv) 2 (sin6  + cos6 ) – 3 (sin4  + cos4) + 1 = 0  cos2 B.cos2 A cos 2 A cos 2 B

sin   sin  cos   cos    0 cos   cos  sin   sin 

QIf cos  + sin  =

sin 4 A  cos4 A sin 2 A  cos2 A

sec   1 sec   1   2 cosec  sec   1 sec   1



3

(ix)

(xv)

(iv)

tan  cot    1  tan   cot   1  sec  cosec  1  cot  1  tan 

1  sin A cos A  1  sin A 1  sin A

(xiii) tan 2 A  tan 2 B 

sin 3   cos3  sin   cos 

(ii) 2 sec2  – sec4  – 2 cosec2  + cosec4  = cot4 – tan4

(v)

2

(iii)

(xvi)

2 cos , show that cos – sin  =

cot A  cosec A  1 1  cos A  cot A – cosec A + 1 sin A

2 sin 

Q20. If sin  + cos  = p and sec  + cosec  = q, show that q ( p 2  1)  2 p . Q21. If x = a sec  + b tan  and y = a tan  + b sec , prove that x2 – y2 = a 2 – b 2. Q22.(i) If sec   x 

1 1 , prove that sec  + tan  = 2x or . 4x 2x

(ii) If sec   tan   p, prove that

Q23. If tan  + sin  = m and tan  – sin  = n, prove that m2  n 2  4 mn . Q24. If sin  + sin2 = 1, prove that cos2+ cos4 = 1 Q25. If 3sin   5cos   5, prove that 5sin   3cos   3 . Q26. If x = a sec  + b tan  and y = a tan  + b sec , prove that x2 – y2 = a 2 – b 2. Q27. If a cos  = x and b cot  = y, show that Q28. If

a2 b2   1. x2 y 2

x y x y x2 y 2 cos   sin   m and sin   cos   n, prove that 2  2  m2  n 2 . a b a b a b

ANSWERS 1. (i)

5 5 , 13 12

6. (i) 1

(ii)

(ii) 1

12 5 , 13 12

8. (i)

11. A = 30°, B = 15° 13. (i) 1 16. 21°

(ii) 2

2. sin θ = 34 2

21 20 21 29 29 , cos θ  , tan θ  , cos ecθ  , sec θ  29 29 20 21 20

(ii)

5 (2  3) 6

(iii) 6 (iv) 9

12. DE = 8 3 cm, AE = 4

1 (iii) 3 17. (i) sin 

15. (i) 1 (ii) 1

3 1 2 2

3 cm

(ii) 2

(iii) 1 – sin  cos 

10.

(iii) 2 (iv)

(iv) 1  2 3

(v) 0

p2 1  sin  . p2  1