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Solving Triangles

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

Sine Rule

         

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 \(\lvert \overline{AC} \rvert^2?\)

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.

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 \[\frac{\sin A + \sin B}{\sin C}?\]

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 \(\left(\sin \angle BAC\right)^2?\)

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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