As the earth revolves around the sun, distant objects can be seen from different locations. This idea can be used to measure the distance between earth and nearby stars and is called stellar parallax. To see how this process works, consider the simplified scenario below.

A.) A distant object is observed on earth from location 1 and again six months later at location 2. The angle at both location 1 and 2 is 89.9 degrees. How many astronomical units is the distant object from location 1?

you have an isosceles triangle with both base angles 89.9°

The base of the triangle is the diameter of earth's orbit.

Knowing that, you can find the altitude of the triangle (the distance to the object).

To determine the distance of a distant object using stellar parallax, we need to measure the apparent shift in its position when observed from different locations on Earth at different times. The angle formed by this shift is known as the stellar parallax angle.

In this scenario, we are given that the angle at both location 1 and location 2 is 89.9 degrees. Since the object is observed six months apart, the angle represents half of the parallax angle. To find the full parallax angle, we can double the measured angle.

So, the parallax angle (p) is given by:

p = 2 * 89.9 degrees

Now, we can use trigonometry to relate the parallax angle to the distance (d) of the object from Location 1. In particular, we can use the small-angle approximation, which states that for small angles in radians, the tangent of an angle is approximately equal to the angle itself.

Using this approximation, we have:

tan(p) = d / 1 AU

Here, 1 AU (Astronomical Unit) represents the average distance between the Earth and the Sun, which is about 149.6 million kilometers or 93 million miles.

Now, we can rearrange the equation to solve for d:

d = tan(p) * 1 AU

To find the value for d, substitute the value for p (which we found to be 2 * 89.9 degrees) and calculate the tangent.

d ≈ tan(2 * 89.9 degrees) * 1 AU

Using a scientific calculator or an online trigonometry calculator, you can find the value of the tangent and calculate the distance.