Problem

Stations \(A\) and \(B\) are \(100~m\) apart, with sight angles \(A~=~40°\) and \(B~=~65°\) to target \(T\), as shown. Find angle \(T\), then use the \(\text{Law}~\text{of}~\text{Sines}\) to find \(BT\) and \(AT\).

Prompt survey triangle with baseline AB=100 m and sight angles A=40° and B=65° to target T; angle T and distances AT and BT are withheld.

Big Picture

What this problem is really about

Model triangulation with the measured baseline and the two sight rays as one complete triangle. Find the remaining angle, then anchor the Law of Sines with the baseline and its opposite angle. Keep each sight distance paired with the angle across from it, and use side-angle order as a final reasonableness check.

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Assume a second 100-m side and use x²=100²+100²-2·100·100cos(40°), giving x≈68.4 m.

Use T=75° but swap opposite pairs, reporting BT≈93.8 m and AT≈66.5 m.

Use T=40°+65°=105°; the sine values produce the same rounded lengths, but the listed angles total 210°, so the triangle record is invalid.

T=180°-40°-65°=75°. Then 100/sin(75°)=BT/sin(40°)=AT/sin(65°), so BT≈66.5 m and AT≈93.8 m.

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Stations \(A\) and \(B\) are \(100~m\) apart, with sight angles \(A~=~40°\) and \(B~=~65°\) to target \(T\), as shown. Find angle \(T\), then use the \(\text{Law}~\text{of}~\text{Sines}\) to find \(BT\) and \(AT\).

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