John K. answered • 04/30/22
Educated and Experienced Math and Science Teacher
The planet, the Sun, and the Earth will always form a triangle. Since we are given two of the angles of this triangle, we could easily fill in the missing angle (by using the Triangle Sum Theorem) if we wanted to. If we do this, and then draw and label a diagram, we will find a right triangle with all angles and the hypotenuse known, but neither of the legs.
Since we have a right triangle with known angles, the missing sides can easily be found using the trigonometric sine and cosine. We can find the distance between the Sun and the planet using the sine of the given angle, and the given distance between the Sun and the Earth.
Incidentally, this is a hypothetical planet, but it is very similar to a real one. There is a real planet that only appears in the sky within about 45° of the sun, meaning that we can only see it over sunrises and sunsets. (It is so bright that sometimes you can find it near the sun during the day, too, but that is extremely difficult.) It looks like a brilliant star hovering over the lovely glow of the fading sun, and for this (probably) it was named after the Roman goddess of beauty. The fact that we can never see it far from the sun in the sky also means that it must orbit the sun more closely than we do. Also, if you like technical terms, the moment when the planet is farthest from the sun in the sky is called "maximum elongation". It is at this moment that the three bodies form a right triangle in space. (For the rest of spring 2022, you can find this planet over the sunrise before dawn.)
