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Solving Triangles

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

Level 3

Find \(PC.\)

NOTE: This length can be expressed as \(\sqrt{\dfrac{A}{B}}\), where \(A,B\) are coprime positive integers. Submit the value of \(A+B\) as your answer.

Aeroplanes A and B are flying with constant velocity in the same vertical plane at angles 30° and 60° with respect to the horizontal, respectively, as shown in the figure.

The airspeed of A is \(100 \sqrt{3} \) meters per second. Initially, an observer in A sees B directly ahead and at the same altitude at a (horizontal) distance of 500 m.

Assuming they take no evasive action and collide, after how many seconds does this happen?

An ant is lost in a square, and his distances to the vertices of the square are 7, 35, 49 and \(x.\) Find \(x.\)

Quadrilateral \(ABCD\) has \(AB=CD,\) and the acute angles \(B\) and \(C\) satisfy \(\sin B = \frac{4}{9}\) and \(\sin C= \frac{5}{6}.\) Find \[\frac{\text{area of }\color{red}{\triangle ABC}}{\text{area of }\color{blue}{\triangle BDC}}.\]

Give your answer to 3 decimal places.

In triangle \( ABC\), \(\angle ACB = 90^\circ \). \(A, D, E,\) and \(B\) are consecutive points on \(AB\) such that \(\overline{AD}=\overline{DE}=\overline{EB}\). If there exists \( \theta\) such that \( \overline{CD} = 5 \cos \theta\) and \(\overline{CE} = 5 \sin \theta \), what is \(\overline{AB}^2?\)

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