The moon forms a right triangle with the earth and the sun during one of its phases, as shown.
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.

if someone is on the moon, and measures the angle from the earth to the sun, then that angle is x (some unknown number). if we let y be some unknown distance from the earth to the sun, and z be another unknown distance from the moon to the sun, then we can tie all of these variables together.

we do so by using the sine ratio which says that sin(angle) = opposite/hypotenuse. in our case, x is the angle in question, y is the opposite or side furthest from angle x, and z is the hypotenuse. the hypotenuse is always the longest side opposite the 90 degree angle.

so using those facts, we can go from sin(angle) = opposite/hypotenuse to the new equation sin(x) = y/z.

now multiply both sides by z

sin(x) = y/z

z*sin(x) = y

then divide both sides by sin(x) to isolate z

z*sin(x) = y

z = y/( sin(x) )

if we knew the numeric values for x and y, then we could compute the actual value of z to find the actual distance from the moon to the sun. since we don't know these values, we leave that last equation as is.

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