The angle of view

Determine the angle of view at which the observer sees a rod 16 m long when it is 18 m from one end and 27 m from the other.

Result

A =  34.919 °

Solution:

a=16 m b=18 m c=27 m  a2=b2+c2−2⋅ b⋅ c⋅ cos⁡A  x=−a2+b2+c22⋅ b⋅ c=−162+182+2722⋅ 18⋅ 27≐797972≐0.82  A=180∘π⋅arccos⁡(x)=180∘π⋅arccos⁡(0.82)≐34.9193≐34.919∘≐34∘55′10"a=16 \ \text{m} \ \\ b=18 \ \text{m} \ \\ c=27 \ \text{m} \ \\ \ \\ a^2=b^2+c^2 - 2 \cdot \ b \cdot \ c \cdot \ \cos A \ \\ \ \\ x=\dfrac{ -a^2+b^2+c^2 }{ 2 \cdot \ b \cdot \ c }=\dfrac{ -16^2+18^2+27^2 }{ 2 \cdot \ 18 \cdot \ 27 } \doteq \dfrac{ 797 }{ 972 } \doteq 0.82 \ \\ \ \\ A=\dfrac{ 180^\circ }{ \pi } \cdot \arccos(x)=\dfrac{ 180^\circ }{ \pi } \cdot \arccos(0.82) \doteq 34.9193 \doteq 34.919 ^\circ \doteq 34^\circ 55'10"

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