Question

A shooting star forms a right triangle with the Earth and the Sun, as shown below:

A right triangle is shown with the vertices labeled Earth, Sun, and Shooting Star. The angle formed by the Sun is labeled x deg

A scientist measures the angle x and the distance y between the Sun and the shooting star. Using complete sentences, explain how the scientist can use only these two measurements to calculate the distance between the Earth and the Sun. (10 points)

Answer 1

- The scientist can use these two measurements to calculate the distance between the Earth and the Sun by applying one of the trigonometric functions: Cosine of an angle.

- The scientist can substitute these measurements into
`cos\alpha=(adjacent)/(hypotenuse)` and solve for the distance between the Earth and the Sun.

Explanation:

Let's assume that the right triangle formed is like the one shown in the figure attached, where "d" represents the distance between the Earth and the Sun.

Then:

The scientist can use only these two measurements to calculate the distance between the Earth and the Sun by applying one of the trigonometric functions: Cosine of an angle.

The scientist can substitute these measurements into
`cos\alpha=(adjacent)/(hypotenuse)`, and solve for the distance "d".

Knowing that:

`\alpha=x\°\\adjacent=d\\hypotenuse=y`

Then:

`cos(x\°)=(d)/(y)`

And solving for "d":

`ycos(x\°)=d`