Ashooting star forms a right triangle with the earth and sun, as shown below

a scientist measured the angle x and the distance y between the earth and the shooting star. using complete sentences, explain how the scientists can use only these two measurements to calculate the distance between the earth and sun.

Yes the scientist can calculate for the distance between the Earth and the Sun by calculating the other angle and then using the sine formula.

SO the first step the scientist that have to do is to calculate for the other angle at the corner of the shooting star. Let’s call the third angle as angle z. We know that in a triangle, the total angle is 180 degrees, therefore:

z = 180 – x – 90

Now we have 2 angles and 1 side. We can now use the sine formula to solve for the distance of Earth and the sun. The sine formula is:

Side 1 / sin (angle opposite to Side 1) = Side 2 / sin (angle opposite to Side 2)

using that:

BC / sin (z) = y / sin (x)

Therefore BC is:

BC = y * sin (z) / sin (x)

The scientist can use only the angle x and the distance y to calculate the distance between the Earth and the Sun using the cosine function.

The cosine of an angle is equal to the ratio of the length of the adjacent side and the length of the hypotenuse. In equation form:

cos x = BC / y

Solving for BC

BC = y cos x

BC is the distance between the Earth and the Sun