So this problem was on our textbook and it seems that the infos given are not enough.

A sector of a circle has an arc of 16.76 cm and with chord 15.43 cm. Find the area of the sector.

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## Comments

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TopNewestYou have \(r\theta = 16.76\) and by cosine rule, \(2r^2 - 2r^2 \cos\theta = 15.43^2\). So you have two equations and two variables, and you're instructed to find \(\dfrac12r^2\theta\).

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im not good at solving variables involving theta

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Do you know numerical/approximation methods?

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From the second equation, you have \(2r^2(1-\cos\theta) =2r^2 \cdot 2\sin^2\left(\dfrac \theta2 \right) = 15.43^2 \Rightarrow 2r \sin \dfrac{\theta}2 = 15.43 \).

\[ (r \theta) \div \left(2r \sin\dfrac\theta2 \right) = \dfrac{16.76}{15.43} \Rightarrow \dfrac{\theta /2}{\sin(\theta /2)} = \dfrac{1676}{1543} \]

Let \(y = \dfrac\theta2 \), then you're left to solve for \(\dfrac y{\sin y} = \dfrac{1676}{1543} \approx 1.08619 \) for positive \(y\) only as \(\theta > 0\) as well.

Now for the numerical methods: Let \(f(y) = \dfrac y{\sin y} \), and we want to find the best estimate of \(y\) which gives \( f(y) \approx 1.08619 \).

Let's try for some value of \(y\):

Comparing all these values of \(f(y) \) against the value \(1.08619\), we can see that the closest value of \(y\) that approximates to \(f(y) \approx 1.08619\) is at \(y = 0.7 \). (Note that the more accurate reading is \(y = 0.698497366525694\ldots \))

Thus we can make a rough estimate that \(y \approx 0.7\) or \(\dfrac \theta2 \approx 0.7 \Rightarrow \theta \approx 1.4 \).

Our desired answer is approximately \( \dfrac{16.76^2}{2\times1.4} = 100\).

Note that the actual value is \(100.536\ldots \).

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