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A triangle $ABC$ is inscribed within the unit circle. Let $x$ be the measure of the angle $C$. Express the length of $AB$ in terms of $x$.

A.) $2\sin x$

B.) $\cos x + \sin x - 1$

C.) $\sqrt{2}(1 - \cos 2x)$

D.) $\sqrt{2}(1 - \sin x)$

I am unsure how to illustrate the circle and calculate the length $x$.

Thank you. `

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  • $\begingroup$ Please read this tutorial about how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig Jun 18 '17 at 8:55
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$$\dfrac{AB}{\sin x}=2R$$ where $R$ is the circum-radius $=1$

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  • $\begingroup$ This solution is based on the Law of Sines. $\endgroup$ – N. F. Taussig Jun 18 '17 at 9:11
  • $\begingroup$ @N.F.Taussig, Thanks for enriching the answer $\endgroup$ – lab bhattacharjee Jun 18 '17 at 9:12
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The angle $AOB=2x $, since it intercepts the same arc than the angle $CAB=x $.

thus by the well-known formula

$$AB^2=OA^2+OB^2$$ $$-2OA.OB.\cos (2x) $$ with

$$OA=OB=radius=1$$

hence

$$AB^2=2 (1-\cos (2x)) $$

$$=4\sin^2 (x) $$ and $$\boxed {AB=2\sin (x) }$$

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