Maths class X : introduction to trigonometry notes

1. In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine

a. sin A, cos A

b. sin C, cos C
2.    Given 15 cot A = 8. Find sin A and sec A 

3.    If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.

4.    In ΔPQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

5.     State whether the following are true or false. Justify your answer. 


a.    The value of tan A is always less than 1..

b.    cos A is the abbreviation used for the cosecant of angle A.

c.    cot A is the product of cot and A

6.    Evaluate the following

a.    sin60° cos30° + sin30° cos 60°

b.    2tan245° + cos230° − sin260°

7.    State whether the following are true or false. Justify your answer.

a.    sin (A + B) = sin A + sin B

b.    The value of sinθ increases as θ increases

c.    The value of cos θ increases as θ increases

d.    sinθ = cos θ for all values of θ

e.    cot A is not defined for A = 0°

8.    Show that 

tan 48° tan 23° tan 42° tan 67° = 1

                      cos 38° cos 52° − sin 38° sin 52° = 0

9.    If tan 2A = cot (A− 18°), where 2A is an acute angle, find the value of A.

10.  If tan A = cot B, prove that A + B = 90°

11.  If sec 4A = cosec (A− 20°), where 4A is an acute angle, find the value of A.

12.  Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

13.  Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

14.  : Write all the other trigonometric ratios of ∠A in terms of sec A.

15.  Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

16.    (sec2q -1 ) (1 - cosec2q )=……………

17.  cot2q–  1/ Sin2q = ............................

18.   Given that sinq =a/b , then cos q is equal to --------------------

19.   If sin q - cos q = 0 , then the value of (sin4q + cos4q) is …………….

20.        Eualuate(1 + cot q - cos q)(1 + tanq + sec q)


21.   If x = a sec q cos Ø  ; y = b sec q sin Ø and z = c tan q , then X2 / a2 + Y2 /b = ……………….

22.   If cosA +cos2 A = 1, then sin2 A + sin2A=

23.   Prove that sec 72/ cos ec18 + sin59/ cos31 = 2 

24.  If sin 2 q = √3 , find q

25.  Prove that cos q - sin q =√ 2 sin q,if sin q + cos q = √2 cos q

26.        Prove that (tanA+  secA- 1) / (tanA-secA + 1) = secA +  tanA

27.   If a cos3 q + 3 cos q sin2q = m a sin3q + 3acos2q sinq = n, 

28.   Prove that(m+ n)2 /3+ (m+ n)2/3= 2a 2 /3

29.        If 1 secq =  x + 1/4x   prove that sec q + tan q = 2x or 1/2x

30.  If √3 tan q = 3 sinq , evaluate sin2q - cos2q

31.        Prove the following identities : 1+ sec A/SecA =  sin2 A/1 - cos A

32.  Prove that :  1/ secq -  tanq      -  1/ cosq  = 1/cosq -1/ secq +  tanq

33.        Prove the following identity:

(sin A + cosec A)2 + ( cos A + sec A )2 = 7 + tan2A + cot2A.

34.        If  x/a cos = y/bsin      and 

ax/cos = by/sin = a2 –b2  prove that x2 /a2 + y2 /b2

35.   If cotA =4/3 check  (1 – tan2A)/ 1 + tan2A   =    cot2A – sin2A

36.   sin (A – B) = ½, cos(A + B) = ½ find A and B

37.   Evaluate tan5° tan25° tan30° tan65° tan85°

38.   Verify  4(sin430° + cos 460°) – 3(cos245° – sin290°) = 2

39.   Show that tan48° tan23° tan 42° tan67° = 1

40.   sec4A = cosec(A – 20) find A

41.   tan A = cot B prove A + B = 90

42.   A, B, and C are the interior angles of DABC show that sin(  B + C  )/2  =   cos  A/2                          

43.   In DABC, if sin (A + B – C) = √3/2 and cos(B + C – A) =1/√2, find A, B and C.

44.  If cos θ =   and θ + φ = 900, find the value of sin φ.

45.  If  tan 2A   =   cot ( A – 180 ),  where  2A  is  an  acute  angle,  find the value of A.

46.  If 2sin (x/2) = 1 , then find the value of x.    

47. If tan A = ½ and tan B = 1/3 , 

by using tan (A + B) =  ( tan A + tan B )/ 1 – tan A. tan B   prove that A + B = 45º

48.   Express sin 76° + cos 63° in terms of trigonometric ratios of angles between 0° and 45°.

49.   Prove that:  2 sec2 θ – sec4 θ – 2 cosec2 θ + cosec4 θ = cot4 θ – tan4 θ  

50.   Find the value of θ for which sin θ – cos θ = 0

51.   Given that sin2A + cos2A = 1, prove that cot2A = cosec2A – 1

52.   If sin (A + B) = 1 and sin (A – B)=1/2  0o< A + B ≤ 90o; A > B, find A and B.

53.   Show that   tan 620/cot 280 =1

54.   If sin A + sin2A = 1, prove that cos2A + cos4A = 1.

55.   If sec 4A = cosec (A – 200), where 4A is an acute angle, find the value of A.

56.   Prove that (cosec θ – sec θ) (cot θ – tan θ) = (cosec θ + sec θ) (sec θ . cosec θ – 2)

57.   Given that A = 60o, verify that 1 + sin A =(Cos A/2  + Sin A/2)2

58.   If sin θ + cos θ = x and sin θ – cos θ = y, show that x2 + y2 = 2

59.   Show that sin4θ – cos4θ = 1 – 2 cos2θ

60.  If θ= 45o. Find the value of sec2θ

61.  Evaluate: cos60 o cos45 o -sin60 o sin45 o

62.  Find the value of tan15 o.tan25 o.tan30 o tan65 o tan85 o

63.  If θ is a positive acute angle such that sec θ = cosec60o, then find the value of 2cos2 θ -1

64.  Find the value of sin65-cos25 without using tables.

65.  If sec5A=cosec(A-36 o). Find the value of A.

66.  If 2 sin x/2  - 1 =0, find the value of x.

67.  If A, B and C are interior angles of ΔABC, then prove that cos (B+C)/2 = sinA/2

68.  Find the value of 9sec2A-9tan2A.

69.  Prove that sin6θ+cos6θ=1-3sin2θcos2θ.

70.  If 5tanθ-4=0, then find the value of (5sinθ - 4cosθ) (5sinθ + 4cosθ)

71.  In ABC, <c=90o, tan A= and tan B=<3.Prove that sin A. cos B+ cos A .sin B=1.

72.  In D ABC, right angled at B, if tan a =1/√3 find the value of Sin A cos C + cos A sin C.

73.  Show that  2(cos4  60 +  sin4 30 )-  (tan2 60  + cot2 45 ) + 3sec2 30 =1/4

74.  sin(50 +q ) - cos(40 -q ) + tan1 tan10 tan 20 tan 70 tan80 tan89 =1

75.  Given tan A =4/3, find the other trigonometric ratios of the angle A.

76.  In a right triangle ABC, right-angled at B, if tan A = 1, then verify that 2 sin A cos A = 1.

77.  In D OPQ, right-angled at P, OP = 7 cm and OQ – PQ = 1 cm. Determine the values of sin Q and cos Q.

78.  In D ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine:(i) sin A , cos A(ii) sin C, cos C

79.  If ÐA and ÐB are acute angles such that cos A = cos B, then show that Ð A = ÐB.

80.  If cot A= 7/8 evaluate: {(1 + sinA )( 1 – sinA)} / {(1+ cosA)(1-cosA)

81.  In triangle ABC, right-angled at B, if tan A = 1/√3   

find the value of :(i) sin A cos C + cos A sin C  (ii) Cos A cos C – sin A sin C

82.  In D ABC, right angled at B, AB = 5 cm and ÐACB = 300 Determine the lengths of the sides BC and AC.

83.  In D PQR, right – angled at Q, PQ = 3 cm and PR = 6 cm. Determine ÐQPR and ÐPRQ

84.  If sin (A-B) = ½ ,cos(A+B ) = ½   A+ B = o < A+ B ≤ 90,  A > B find A  and  B

85.  Evaluate the following: (5cos260 + 4sec230 -  tan2 45)/ (sin2 30 + cos2 30)

86.  If sin 3 A = cos (A – 26), where 3 A is an acute angle, find the value of A.

87.  Prove the trigonometric identities   (1 - cos A)/( 1 – cos A) = (cosec A – cot A)2           

88.  Prove the trigonometric identities ( 1+ 1/tan2A) (1 + 1/cot2A) =  1/(sin2A- cos4A)

89.  Prove the trigonometric identities  (sec4A – sec2A) = tan4A +tan2A = sec 2 A tan2  A

90.  Prove the trigonometric identities  cotA – tanA = (2cos 2A -1)/ (sinA.cosA)

91.  Prove the trigonometric identities. (1- sinA +cosA)2 = 2(1+cosA )(1 – sinA)

92.  If tanA +sinA = m and tanA – sinA=n show that m2 – n2 = 4 

93.   If x= psecA + qtanA  and  y= ptan A +q secA prove that x2 – y2 = p2 – q2

94.  If sinA + sin2A = 1   prove that cos2 A  + cos4 A =1

95. Express the following in terms of t-ratios of angles between 0° and 45°.

1)    sin 85° +cosec 85°

2)    cosec 69° +cot 69°

3)    sin 81° +tan 81°

4)    cos 56° +cot 56°

96. [sin (90 -A) sin A]/tan A-1 = - sin² A

97. cos  cos(90° -  ) -sin  sin (90° -  ) = 0

98. sin (90° -  ) cos (90° -  ) = tan  /(1 +tan²  )

99. cosec² (90° -  ) -tan²    = cos²(90° -  ) +cot²  

100.  If cos /cos  = m and cos /sin  = n, show that (m² +n²) cos²  = n².If x = r cos  sin , y = r cos  cos  and z = r sin  , show that x² +y² +z² = r².