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Angle ABC and vertical angle P-Q (2062)

Short answer

Horizontal angle ABC: BA mean -0°00′20″, BC mean 160°20′15″, difference 160°20′35″. Vertical: FL = 61°30′00″, FR = 298°30′30″, index error +15″, zenith distance 61°29′45″, vertical angle = 28°30′15″ elevation.

Find the horizontal angle ABC from the following field observations and also the vertical angle from P to Q. [Figure: at B, line to A observed FR 179°59′20″ and FL 00°00′00″; line to C observed FL 160°20′30″ and FR 340°20′00″. Find horizontal angle ABC.] [Figure: instrument at P sighting to Q, observed FL 61°30′00″ and FR 298°30′30″. Find vertical angle from P to Q.]

(a) Horizontal angle ABC (instrument at B)

Mean direction = [FL + (FR - 180°)]/2.

BA:  FL=0°00′00″,  FR=179°59′20″⇒mean=359°59′40″ (=−0°00′20″)BC:  FL=160°20′30″,  FR=340°20′00″⇒mean=160°20′30″+160°20′00″2=160°20′15″∠ABC=160°20′15″−(−0°00′20″)=160°20′35″\begin{aligned} \text{BA}: &\; FL = 0°00′00″,\; FR = 179°59′20″ \Rightarrow \text{mean} = 359°59′40″\ (= -0°00′20″)\\ \text{BC}: &\; FL = 160°20′30″,\; FR = 340°20′00″ \Rightarrow \text{mean} = \frac{160°20′30″ + 160°20′00″}{2} = 160°20′15″\\ \angle ABC &= 160°20′15″ - (-0°00′20″) = 160°20′35″ \end{aligned}

(b) Vertical angle P to Q (instrument at P)

FL = 61°30′00″, FR = 298°30′30″ (zenith-angle circle).

Index error=(FL+FR)−360°2=360°00′30″−360°2=0°00′15″Z=FL−FR+360°2=61°30′00″−298°30′30″+360°2=61°29′45″Vertical angle=90°−Z=28°30′15″\begin{aligned} \text{Index error} &= \frac{(FL+FR) - 360°}{2} = \frac{360°00′30″ - 360°}{2} = 0°00′15″\\ Z &= \frac{FL - FR + 360°}{2} = \frac{61°30′00″ - 298°30′30″ + 360°}{2} = 61°29′45″\\ \text{Vertical angle} &= 90° - Z = 28°30′15″ \end{aligned}

Answer: horizontal angle ABC = 160°20′35″; vertical angle from P to Q = 28°30′15″ (elevation).

Note: a zenith-type vertical circle is assumed.

✓ Answer checked by Claude Opus 5.5 (high effort)

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