To solve this problem, we will use the trigonometric function tangent. Recall that by definition in a right triangle:
[tex]\tan \theta=\frac{opposite\text{ leg}}{\text{adjacent leg}}\text{.}[/tex]In the given diagram, opposite leg=1 A.U. ( astronomical unit), θ=0.00001389°, and adjacent leg =d, therefore:
[tex]\tan (0.00001389)=\frac{1}{d}\text{.}[/tex]Solving the above equation for d, we get:
[tex]d=\frac{1A.U\text{.}}{\text{tan(}0.00001389)}\text{.}[/tex]Simplifying, we get:
[tex]d=4124966.128\text{ A.U.}[/tex]Answer:
[tex]d=4124966.128\text{ A.U.}[/tex]For any star, if we are given an angle θ, we can use the following expression to determine its distance from the Sun:
[tex]d=\frac{1A.U\text{.}}{\tan\theta}[/tex]