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

If the angles of a triangle are in ratio 4:1:1, then the ratio of longest side to perimeter is:

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In the figure, \(\triangle ABC\) is an equilateral triangle where \(AB = 6\) cm. The area of the circumcircle is \(k\pi\) cm\(^{2}\). Find the value of \(k\).

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If \(CD=\sqrt{3},\) \(PQ=2,\) and \(\angle PDQ=30^\circ,\) find \(CP.\)

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In triangle \(ABC\), \( \angle BAC = 2 \angle ABC \), \( AB= 31, AC = 13 \). What is \(BC^2\)?

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If \( AP = AC + 2\cdot CB \), what is the measure \(\angle ABP \) in degrees?

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