The triangle \(PTA\) is shown. The length of \(PA\) is 75 m and the length of \(PT\) is 51 m.
The angle of depression from \(T\) to \(A\) is 36°, and the angle \(PTA\) is obtuse.
Find the length of \(TA\). Give your answer correct to 2 decimal places. (3 marks)
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\(\angle TAP=36^{\circ} \ \text {(alternate)}\)
\(\text{Using sine rule in} \ \triangle TAP:\)
| \(\dfrac{\sin \angle PTA}{75}\) | \(=\dfrac{\sin 36^{\circ}}{51}\) |
| \(\sin \angle PTA\) | \(=75 \times \dfrac{\sin 36^{\circ}}{54}=0.864 \ldots\) |
| \(\angle PTA\) | \(=\sin ^{-1}(0.864 \ldots)=180-59.81=120.19^{\circ}\ \ \text{(obtuse)}\) |
\(\angle PTX=120.19-54=66.19^{\circ}\)
\(\angle TPA=90-66.19=23.81^{\circ}\ \left(180^{\circ}\ \text{in}\ \triangle \right)\)
\(\text{Using sine rule in} \ \triangle TAP:\)
| \(\dfrac{TA}{\sin 23.81^{\circ}}\) | \(=\dfrac{51}{\sin 36^{\circ}}\) |
| \(TA\) | \(=\dfrac{51 \times \sin 23.81^{\circ}}{\sin 36^{\circ}}\) |
| \(=35.03 \ \text{m (2 d.p.)}\) |