Astronomers often measure large distances using astronomical units (AU) where 1 AU is the average distance from Earth to the Sun. In the image d represents the distance from a star to the Sun.Using a technique called "stellar parallax" astronomers determined 6 is 0.00001389 degrees. square o How far away is the star from the Sun in astronomical units? To the nearest thousand astronomical units, the star is square AU units away from the Sun.

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Astronomers often measure large distances using astronomical units (AU) where 1 AU is the average distance from Earth to the Sun. In the image d represents the distance from a star to the Sun.Using a technique called "stellar parallax" astronomers determined 6 is 0.00001389 degrees. square o How far away is the star from the Sun in astronomical units? To the nearest thousand astronomical units, the star is square AU units away from the Sun.

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To solve this problem, we can use trigonometry. The angle \(\theta\) given is the angle at Earth, and we have the lengths of the sides of the triangle in terms of astronomical units (AU). The side opposite the Sun is 1 AU, which is the baseline for the parallax measurement, and the side opposite the star is labeled as \(d\), which is what we want to find.The tangent of the angle \(\theta\) in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is 1 AU and the adjacent side is \(d\). Therefore, we can write:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{1 \text{ AU}}{d}\]We can rearrange this to solve for \(d\):\[d = \frac{1 \text{ AU}}{\tan(\theta)}\]Given that \(\theta\) is 0.00001389 degrees, we need to convert this angle to radians because the tangent function in most calculators requires the angle to be in radians. To convert degrees to radians, we use the conversion factor \(\pi\) radians = 180 degrees:\[\theta \text{ in radians} = \theta \text{ in degrees} \times \frac{\pi}{180}\]\[\theta \text{ in radians} = 0.00001389 \times \frac{\pi}{180}\]Now, let's calculate \(\theta\) in radians:\[\theta \text{ in radians} = 0.00001389 \times \frac{3.141592653589793}{180}\]\[\theta \text{ in radians} \approx 0.00001389 \times \frac{3.141592653589793}{180}\]\[\theta \text{ in radians} \approx 2.42406840554768 \times 10^{-7}\]Now we can calculate \(d\):\[d = \frac{1 \text{ AU}}{\tan(2.42406840554768 \times 10^{-7})}\]Using a calculator to find the tangent of \(\theta\) in radians:\[d \approx \frac{1 \text{ AU}}{\tan(2.42406840554768 \times 10^{-7})}\]\[d \approx \frac{1 \text{ AU}}{2.42406840554768 \times 10^{-7}}\]\[d \approx \frac{1}{2.42406840554768 \times 10^{-7}}\]\[d \approx 4.12310562561766 \times 10^{6} \text{ AU}\]Rounding to the nearest thousand astronomical units:\[d \approx 4,123,000 \text{ AU}\]So, to the nearest thousand astronomical units, the star is \(\boxed{4,123,000}\) AU units away from the Sun.

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