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Chapter 8: Q. 55 (page 537)

Show that the length d of a chord of a circle of radius r is given by the formula d=2rsin胃2 where u is the central angle formed by the radii to the ends of the chord. See the figure. Use this result to derive the fact that sin胃<胃, where 胃>0is measured in radians.

Short Answer

Expert verified

The result d=2rsin胃2has been proved.

And it is used to derive the fact that sin胃<胃sin胃<胃

Step by step solution

01

Step 1. Given Information

The chord with center O and radius r.

sin胃<0,胃>0

02

Step 2. Use law of cosines in ∆OAB

By using Law of cosines,

(AB)2=(OA)2+(OB)2-2(OA)(OB)鈥�cos胃d2=r2+r2-2r2cos胃=4r2(1-cos胃2)d=2r1-2sin2(胃2)2=2rsin胃2

Hence the first result has been proved.

03

Step 3. Find the angle

Angle=ArclengthRadius

胃=arclengthofABrAB=r胃

in which 胃is in radians.

04

Step 4.

Since length <r胃AB<arclengthAB

d<r胃2rsin胃2<胃2sin胃2<胃2

05

Step 5. Second result

Since 胃is an angle interior angle of a triangle,

0<胃<蟺0<胃2<蟺2

cos胃2<1cos胃2.sin胃2<胃2as0<胃<蟺0<sin胃2<12cos胃2sin胃2<胃sin胃<胃

which is the required result.

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