   Chapter 3.8, Problem 38E

Chapter
Section
Textbook Problem

# In the figure, the length of the chord AB is 4 cm and the length of the arc AB is 5 cm. Find the central angle θ , in radians, correct to four decimal places. Then give the answer to the nearest degree. To determine

To find:

Central angle θ

Explanation

1) Concept:

Working rule of Newton’s method is,

Start with initial approximation x1  which is obtained from graph. Evaluate fx1and f'(x1) and apply the Newton’s formula. After the first iteration we get x2. Repeat the process with x1  replaced by x2. Calculate the approximations x1, x2, x3,  until the successive approximations xn, xn+1  agree the four decimal places.

2) Formula:

i) Length of arc (S)

s=rθ

ii) Law of cosines

It is related to the triangle,

a2=b2+c2-2·b·c·cosA

iii) Distributive property:

ab-c=ab-ac

iv) Power rule of differentiation:

ddxxn=nxn-1

3) Given:

Length of chord AB = 4 cm

Length of arc AB = 5cm

4) Calculation:

By length of arc formula,

s=rθ

Substituting the values,

5=rθ

r=5θ

Let C be the center of circle as shown on figure. So, AC=BC=r

By using cosine rule for  ABC ,

c2=a2+b2-2·a·b·cosθ

42=r2+r2-2·r·r·cosθ

42=2r2-2r2·cosθ

42=2r2(1-cosθ)

16=25θ2(1-cosθ)

16=225θ2(1-cosθ)

Multiplying by θ2

16θ2=50(1-cosθ)

By use of distributive property,

16θ2=50-50cosθ

To solve this equation,

Let fθ= 16θ2-50+50cosθ

We have to use Newton’s method for this function, so

Differentiating fθ with respect to θ,

Now by power rule and ddθcos θ=-sinθ,

f'θ= 32θ-50sinθ

Using a graphing calculator, sketch the graph of fθ to find the initial point

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