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Trigonometric problem solving culminates in this chapter. Leave no side and no angle unmeasured!

In triangle \(ABC\) above, \[\angle BAC =x= 30^{\circ}, \sin \angle ABC = \frac{1}{4}, \lvert \overline{BC} \rvert =a= 9,\] where \(\lvert \overline{BC} \rvert\) is the length of \(\overline{BC}.\) What is the value of \(\lvert \overline{AC} \rvert^2?\)

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Triangle \(ABC\) has angles \(\angle A = 60^{\circ}\) and \( \angle B = 45^{\circ} \), and a side length \(a = 21 \sqrt{6}\). What is the side length \(b\)?

**Details and assumptions**

\(a\) and \(b\) are the lengths of the sides opposite to the vertices \(A\) and \(B\), respectively.

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In triangle \(ABC\) above, we are given the side lengths as follows:\[\lvert\overline{BC}\rvert=a, \lvert\overline{CA}\rvert=b, \lvert\overline{AB}\rvert=c.\] If \(ab : bc : ca = 2 : 11 : 5,\) what is the value of \[\frac{\sin A + \sin B}{\sin C}?\]

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In triangle \(ABC\) above, \(\angle ABC =x= 30^{\circ}\) and \[\lvert \overline{BC} \rvert =a= 6, \lvert \overline{CA} \rvert =b= 10.\] What is the value of \(\left(\sin \angle BAC\right)^2?\)

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In triangle \(ABC\) above, we are given the side lengths as follows:\[\lvert\overline{BC}\rvert=a, \lvert\overline{CA}\rvert=b, \lvert\overline{AB}\rvert=c.\] If \(\sin A : \sin B = 5 : 4\) and \(\ a : c = 2 : 5,\) what is \(a : b : c?\)

**Note:** The above diagram is not drawn to scale.

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