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.

Q2. Given 20cotθ21

find all other trigonometric ratios.

Q3. If 12cos A13

verify that: 35sin A(1 tan A)156

.

Q4. (i) If 7 cot 24, prove that 1 cos 11 cos 7

(ii) If 4cot 5 , show that: 5sin 3cos 75sin 2cos 2

.

Q6. If 21cosec = 29, find the value of : (i) 2 2

2

cos sin1 2sin

(ii)2

2 2

2cos 1cos sin

Q7. If 1tan 2;tan

show that: 22

1tan 2tan

.

Q8. Evaluate each of the following :

(i) 2 cos2 60 cot 30° + 6 sin2 30° cosec2 60° (ii) 2 2 25sin 30 cos 45 4 tan 30

2sin 30 cos30 tan 45

(iii) 2 (cos2 45° + tan2 60°) – 6 (sin245° – tan230°) (iv) 2 2 2 2

2

tan 60 3sec 30 4cos 45 5cos 90cosec 30 sec60 cot 30

Q9. If = 30°, verify that : (i) 2

2 tansin 21 tan

(ii)

2

2

1 tancos 21 tan

(iii) 2

2 tantan 21 tan

Q10. Given that sin (A + B) = sin A cos B + cos A sin B, find the value of sin 75°.

Q11. If 3sin (A 2B)2

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

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.

E B

C

A

D 20 cm

12 c

m

30°

Q13. Evaluate each of the following:

(i) 2 2

2 2

cos 20 cos 70sin 57 sin 33

(ii) 2 2sin 27 cos 63

cos63 sin 27

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

Q14. Prove that:

(i) sin .cos (90 cos cos sin (90 ).sin 1sin (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) 2 2

2 2

cos 40 cos 50cos (40 ) sin (50 )sin 40 sin 50

(ii) 2 2 sin15 cos75 cos15 sin 75sec 10 cot 80cos sin(90 ) sin . cos (90 )

(iii) 2 2tan cot (90 ) sec cosec (90 ) sin 35 sin 55

tan10 tan 20 tan 45 tan 70 tan80

(iv) 2 2tan 20 cot 20 2 tan15 tan 37 tan53 tan 60 tan 75

cosec70 sec70

(v) 2 2sec39 2 .tan17 tan38 tan 60 tan 52 tan 73 3(sin 31 sin 59 )cosec 51 3

Q16. If sec 5θ = cosec (θ – 36°), where 5θ is an acute angle, find the value ofθ . Q17. Simplify the following expressions:

(i) (1 + cos ) (cosec – cot ) (ii) cosec (1+ cos ) (cosec – cot ) (iii) 3 3sin cos

sin cos

(iv) 4 4

2 2

sin cossin cos

A AA A

Q18. Prove that following identities: (i) cosec2 + sec2 = cosec2 . sec2 (ii) 2 sec2 – sec4 – 2 cosec2 + cosec4 = cot4 – tan4

iii) 22

2

1 tan 1 tan1 cot1 cot

(iv) tan cot 1 tan cot 1 sec cosec

1 cot 1 tan

(v) 1 sin cos1 sin 1 sin

A AA A

(vi) sec 1 sec 1 2 cosec

sec 1 sec 1

(vii) sin sin2cot cosec cot cosec

(viii) 2 2 2 4

1 1 11 1tan A cot A sin A sin A

(ix) 2 3sin cos 1 sin cos

1 cot cos sin

(x) 2 3cos sin 1 sin cos

1 tan sin cos

(xi) 21 cos sin cot

sin (1 cos )

(xii) 1 1 1 1cosec A cot A sin A sin A c osec A cot A

(xiii) 2 2 2 2

2 22 2 2 2

cos B cos A sin A sin Btan A tan Bcos B.cos A cos A cos B

(xiv) 2 (sin6 + cos6 ) – 3 (sin4 + cos4) + 1 = 0

(xv) sin sin cos cos 0cos cos sin sin

(xvi) cot A cosec A 1 1 cos A

cot A – cosec A + 1 sin A

QIf cos + sin = 2 cos , show that cos – sin = 2 sin Q20. If sin + cos = p and sec + cosec = q, show that 2( 1) 2q p p .

Q21. If x = a sec + b tan and y = a tan + b sec , prove that x2 – y2 = a2 – b2.

Q22.(i) If 1sec ,4

xx

prove that sec + tan = 2x or 12x

. (ii) If sec tan ,p prove that 2

2

1 sin1

pp

.

Q23. If tan + sin = m and tan – sin = n, prove that 2 2 4m n 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 = a2 – b2.

Q27. If a cos = x and b cot = y, show that 2 2

2 2 1a bx y

.

Q28. If cos sinx y ma b

and sin cos ,x y na b

prove that 2 2

2 22 2 .x y m n

a b

ANSWERS

1. (i) 5 5,13 12

(ii) 12 5,13 12

2. 21 20 21 29 29sinθ= ,cosθ , tan θ ,cos θ ,secθ29 29 20 21 20

ec

6. (i) 1 (ii) 1 8. (i) 3 42 (ii) 5 (2 3)

6 (iii) 6 (iv) 9 10. 3 1

2 2

11. A = 30°, B = 15° 12. DE = 8 3 cm, AE = 4 3 cm

13. (i) 1 (ii) 2 (iii) 13

15. (i) 1 (ii) 2 (iii) 2 (iv) 1 2 3 (v) 0

16. 21° 17. (i) sin (ii) 1 (iii) 1 – sin cos (iv)