Question

HELP PLEASE! Geometry! The moon forms a right triangle with the Earth and the sun during one of its phases, as shown below.

A right angle triangle is shown with the Earth at the right angle. The acute angle between the line joining the Earth and the moon and the sun and the moon is x degrees. The distance between the Earth and the sun is y.

A scientist measures the angle x and the distance y between the Earth and the sun. Using complete sentences, explain how the scientist can use only these two measurements to calculate the distance between the moon and the sun.

Answer 1

Use trig. functions:
Are you familiar with the trigonometry expression: SOH CAH TOA?

Answer 2

@philo1234 Yes :) I was thinking maybe it would be tan?

Answer 3

You can use any of them. Let's use Sine expression:
\[Sin \theta = \frac{ opposite }{ hypotenuse }\]

Angle = x
opposite side to the angle 'x' is 'y'.
Can you make a statement for the hypotenuse (distance between the moon and the sun)

Answer 4

I'm not sure what I could say about the hypotenuse. Maybe it's unknown? or undefined? Or y is adjacent to it?

Answer 5

The hypotenuse represents the distance between the moon and the sun.
The first statement using sine:
The distance between the earth and the sun can be found by multiplying the sine of the angle x and the distance between the earth and the sun (y).

Answer 6

\[Cos \theta = \frac{ adjacent }{ hypotenuse }\]

Answer 7

Let us use the angle between the earth to the sun to the moon: (180-(90+x)=90-x).
Using Cosine:
Statement:
The distance between the earth and the sun can be found by multiplying the cosine of the angle 90-x and the distance between the earth and the sun (y).

Answer 8

Do you understand the statements and how I got them?
Statement 1:
The distance between the earth and the sun can be found by multiplying the sine of the angle x and the distance between the earth and the sun (y).

Statement 2:
The distance between the earth and the sun can be found by multiplying the cosine of the angle 90-x and the distance between the earth and the sun (y).

Answer 9

Yep :) I was thinking about SOH-CAH-TOA in my head and I could see where you got the ratios. The 2nd statement was hard for me to understand at first but I'm working it out in my head. Thank you so much! You are a life saver & it made a lot of sense. I understand now, thank you very much :) @philo1234

Answer 10

Hold on I made a mental error here are the revise statements:
Statement 1:
The distance between the earth and the sun can be found by dividing the distance between the earth and the sun (y) by the sine of the angle x. Statement 2: The distance between the earth and the sun can be found by dividing the distance between the earth and the sun (y) by the cosine of the angle 90-x.

Answer 11

@dellzasaur, Please look at my post before this.