trigonometry exercise 22b

Sol. In right MBC, LB= 90° 3 AC = 20 cm tan A = - ) 4 BC 3 3x But tan A = = - = - AB 4 4x :. BC= 3 x cm and AB= 4 x cm No\v AC 2 = AB 2 + BC 2 ::::> (20) 2 = ( 4 x) 2 + (3 x) 2 ::::> 400 = 16 x 2 + 9 x 2 = 25 x 2 x2 = 400 = 16 = (4)2 25 . • • . . . x=4 Hence AB = 4 x = 4 x 4 = 16 cm and BC = 3 x = 3 x 4 = 12 cm Ans. 2x Q. 22. If cos 8 = 2 , find the values of l+x sin 8 and tan 8 in tenns of X. 2 x Base Sol. cos 9 = 2 - H I+ x yp. . . . AB -- AC AB=2x and AC= I +x 2 No\v in right MBC, AC 2 = AB 2 + BC 2 c (Pythagoras Theoren1) =>(I + x 2 ) 2 = (2 x) 2 + BC 2 ::::> 1 + 2 x 2 + x 4 = 4 x 2 + BC 2 ::::> BC 2 = 1 + 2 x 2 + x 4 - 4 x 2 = 1 - 2 x 2 + x 4 => BC 2 = (1 -x 2 ) 2 ::::> BC 2 = (x 2 - 1) 2 :. BC= x 2 - 1 . e Perp. x 2 - I x 2 -1 Now sin = Hyp. - 1 + x2 - x2 + 1 e Perp. (x 2 - 1) tan = = A Base 2 x ns. EXEiCISE 22 (B) Q. 1. Without using trigonometric tables, find the values of : (i) sin 60° cos 30° +cos 60° sin 30° (ii) sin 45° cos 30° - cos 45° sin 30° (iii) cos 60° cos 45° + sin 60° sin 45° (iv) cos 90° + cos 2 45° sin 30° tan 45° Sol. (i) sin 60° cos 30° +cos 60° sin 30° J3 J3 1 I --x-+-x- 2 2 2 2 3 I 4 = 4 + 4 = 4 = I Ans. (ii) sin 45° cos 30° - cos 45° sin 30° l J3 I 1 J3 1 = J2 x 2 - J2 x 2 = 2J2 - 2J2 J3 -1 = 2 J2 Ans. (iii) cos 60° cos 45° + sin 60° sin 45° 1 I J3 I 1 J3 = 2 x J2 + 2 x J2 = 2J2 + 2J2 I+ J3· J3 +I , = 2 J2 = 2 J2 Ans. (iv) cos 90° + cos 2 45° sin 30° tan 45°

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Sol. In right MBC, LB= 90

3 AC = 20 cm tan A = -) 4 BC 3 3x

But tan A = = - = -AB 4 4x

:. BC= 3 x cm and AB= 4 x cm No\v AC2 = AB2 + BC2

::::> (20)2 = ( 4 x)2 + (3 x)2 ::::> 400 = 16 x2 + 9 x2 = 25 x2

x2 = 400 = 16 = (4)2 25

.

.

. . x=4 Hence AB = 4 x = 4 x 4 = 16 cm and BC = 3 x = 3 x 4 = 12 cm Ans.

2x Q. 22. If cos 8 = 2 , find the values of l+x sin 8 and tan 8 in tenns of X.

2 x Base Sol. cos 9 = 2 - H I+ x yp.

.

. .

AB --

AC AB=2x

and AC= I +x2

No\v in right MBC, AC2 = AB2 + BC2

c

(Pythagoras Theoren1) =>(I + x2) 2 = (2 x)2 + BC2

::::> 1 + 2 x2 + x4 = 4 x2 + BC2

::::> BC2 = 1 + 2 x2 + x4 - 4 x2

= 1 - 2 x2 + x4

=> BC2 = (1 -x2)2 ::::> BC2 = (x2 - 1)2 :. BC= x2 - 1

. e Perp. x 2 - I x 2 -1 Now sin = Hyp. - 1 + x2 - x2 + 1

e Perp. (x2

- 1) tan = = A Base 2 x ns.

EXEiCISE 22 (B) Q. 1. Without using trigonometric tables, find

the values of : (i) sin 60 cos 30 +cos 60 sin 30

(ii) sin 45 cos 30 - cos 45 sin 30 (iii) cos 60 cos 45 + sin 60 sin 45 (iv) cos 90 + cos2 45 sin 30 tan 45 Sol. (i) sin 60 cos 30 +cos 60 sin 30

J3 J3 1 I --x-+-x-

2 2 2 2 3 I 4

= 4 + 4 = 4 = I Ans. (ii) sin 45 cos 30 - cos 45 sin 30

l J3 I 1 J3 1 = J2 x 2 - J2 x 2 = 2J2 - 2J2

J3 -1 = 2J2 Ans.

(iii) cos 60 cos 45 + sin 60 sin 45 1 I J3 I 1 J3

= 2 x J2 + 2 x J2 = 2J2 + 2J2 I+ J3 J3 +I ,

= 2J2 = 2J2 Ans. (iv) cos 90 + cos2 45 sin 30 tan 45