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Chapter 8: Q. 44 (page 542)

Refer to the figure, in which a unit circle is drawn. The line segment DB is tangent to the circle and 胃 is acute.

(a) Express the area of 鈭�OBCin terms of sin胃 and cos胃.
(b) Express the area of 鈭�OBDin terms of sin胃 and cos胃.

(c) The area of the sector OBC铃�of the circle is 12胃, where 胃is measured in radians. Use the results of parts (a) and (b) and the fact that

Area of 鈭�OBC<Area of OBC铃�<Area of 鈭�OBDto show that,

1<胃sin胃<1cos胃

Short Answer

Expert verified

(a) The area of 鈭�OBCis sin胃2

(b) The area of 鈭�OBDis sin胃2cos胃

(c) The area of the sectorOBC铃� is 12胃

Step by step solution

01

Part (a) Step 1. Given Information

The line segment DB is tangent to the circle

胃 is acute.

02

Part (a) Step 2. Consider the figure

Consider the figure:

Radius, OB=OC=1

鈭�COB=胃

The line segment DB is tangent to B

鈭�OBD=90掳

03

Part (a) Step 3. Consider ∆OBC

Area of 鈭�OBC

鈭�OBC=12(OB)(OC)sin(鈭�COB)=12(1)(1)sin胃=sin胃2

04

Part (b) Step 1. Find the area ∆OBD

Area of tan胃=DBOBDB=tan胃(sinceOB=1)

Area of 鈭�OBD=12bh

=12脳DB脳OB=12脳tan胃脳1=sin胃2cos胃

05

Part (c) Step 1. Find area of OBC⏜

Areaof鈭�OBC<AreaofOBC铃�<Areaof鈭�OBD

sin胃2<胃2<sin胃2cos胃sin胃<胃<sin胃cos胃1<胃sin胃<1cos胃

Hence the required result.

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