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.

To solve this problem you must apply the proccedure shown below:

 1- You have the following information given in the problem above:

 - The moon forms a right triangle with the Earth and the Sun during one of its phases.

 - The 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.

 2- Therefore, you have:

 d: distance between the moon and the sun

 Sin(α)=opposite/hypotenuse

 α=x
 opposite=y
 hypotenuse=d

 3- Therefore, you have:

 Sin(x)=y/d
 d=ySin(x)

 The answer is: The scientist can calculate the distance between the moon and the sun, by applying the trigonometric identity: Sinα=opposite/hypotenuse, where the opposite side is the distance between the earth and the sun (y), α is the acute angle between the line joining the Earth and the moon and the Sun and the moon (x degrees) and the hypotenuse is the distance that they need to calculate (d). Therefore, the need to solve for d.

(-14+3/2b) - (1+8/2b) = -5 -5/2b(-10+b) + (7b-5) = 8b-15(7/2b-3) - (8+6b) = -5/2b-11(5+2b) + (2b+3/2) = 4b+ 13/2