Question

Astronomers often measure large distances using astronomical units (AU) where 1 AU is the average distance from Earth to the Sun. In the image, drepresents the distance from a star to the Sun. Using a technique called "stellar parallax," astronomers determined 0 is 0.00001389 degrees.
write an expression to calculate d for any star

Answer 1

By calculating the distance of a star to the Sun, astronomers can calculate the distance from Earth to the star.

• $\displaystyle \mathrm{The \ \mathbf{expression} \ to \ calculate \ \mathit{d} \ for \ any \ \mathbf{star} \ is; \ \underline{d \ is \ \frac{1 \, AU}{tan(\theta)}}}$

Reasons

The given parameters are;

The distance from the star to the Sun = d

The angle formed by the ray to the Sun and the ray to Earth from the star, θ = 0.00001389 degrees

The average distance from Earth to the Sun = 1 AU

By trigonometric ratios, we have;

$\displaystyle tan(\theta) = \mathbf{\frac{Opposite}{Adjacent}}$

Therefore;

$\displaystyle tan(\theta) = \frac{1}{d}$

The expression to calculate for d for any star is therefore;

• $\displaystyle \underline{ d \ is \ \frac{1 \, AU}{tan(\theta) }}$

The angle given for θ for the referenced star is θ = 0.00001389

Therefore;

$\displaystyle d = \frac{1 \, AU}{tan(0.00001389^{\circ}) } \approx 4,124,966.13 \, AU$

Learn more about trigonometric ratio here:

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