In the tenth century, the mathematician 'Abd al-'Aziz...

In the tenth century, the mathematician 'Abd al-'Aziz alQabisi described a trigonometric method, using only the sine, for determining the height and distance of an inaccessible object. One sights the summit $A$ from two locations $C$, $D$, and determines, using an astrolabe (an angle-measuring instrument usually used for astronomical purposes), the angles $\alpha_{1}=\angle A C B$ and $\alpha_{2}=\angle A D B$ (Fig. 9.37). If $C D=$ $d$, then the height $y=A B$ and the distance $x=B C$ are given by $$ \begin{aligned} &y=\frac{d \sin \alpha_{2}}{\sin \left(90-\alpha_{2}\right)-\frac{\sin \left(90-\alpha_{1}\right) \sin \alpha_{2}}{\sin \alpha_{1}}} \\ &x=\frac{y \sin \left(90-\alpha_{1}\right)}{\sin \alpha_{1}} \end{aligned} $$ Prove that al-Qabisi's formula is correct. Al-Qabi's method for determining height and distance by way of two angle determinations

In the tenth century, the mathematician 'Abd al-'Aziz alQabisi described a trigonometric method, using only the sine, for determining the height and distance of an inaccessible object. One sights the summit $A$ from two locations $C$, $D$, and determines, using an astrolabe (an angle-measuring instrument usually used for astronomical purposes), the angles $\alpha_{1}=\angle A C B$ and $\alpha_{2}=\angle A D B$ (Fig. 9.37). If $C D=$ $d$, then the height $y=A B$ and the distance $x=B C$ are given by
$$
\begin{aligned}
&y=\frac{d \sin \alpha_{2}}{\sin \left(90-\alpha_{2}\right)-\frac{\sin \left(90-\alpha_{1}\right) \sin \alpha_{2}}{\sin \alpha_{1}}} \\
&x=\frac{y \sin \left(90-\alpha_{1}\right)}{\sin \alpha_{1}}
\end{aligned}
$$
Prove that al-Qabisi's formula is correct.
Al-Qabi's method for determining height and distance by way of two angle determinations

Key Concepts

Trigonometric Ratios

Trigonometric ratios express the relationship between the angles and sides of a right-angled triangle. They form the basis for connecting an angle's measure to the ratio of two of its sides (for example, sine, cosine, and tangent), which is essential in solving problems involving unknown distances or heights by using known angles and side lengths.

Law of Sines

The Law of Sines relates the lengths of the sides of any triangle to the sines of its angles. It provides a crucial method for solving non-right triangles, allowing one to determine unknown side lengths given some known angles and one side length, making it a powerful tool in proving and deriving formulas for height and distance measurements in surveying applications.

Angle of Elevation

An angle of elevation is the angle formed by the horizontal line of sight and the line of sight up to an object. This concept is essential in indirect measurements of height, since by measuring this angle from a fixed distance or along a baseline, one can apply trigonometric relationships to determine the object's height without needing to measure it directly.

Trigonometric Methods in Indirect Measurement

Trigonometric methods in indirect measurement involve using observed angles and known distances to calculate unknown heights and distances. This approach is widely applied in surveying and astronomy, where direct measurement is often impractical, and relies on constructing triangles where the relationships between angles and side lengths, as captured by trigonometric identities and the Law of Sines, can be exploited to establish the desired quantities.

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