Explanation:
hey, you just changed the angles in the question.
this following answer was about the angles of depression of 15° and 30°.
you cannot change the problem, when the answers are already given for the original problem.
so, I will add a copy with the adapted numbers for 45° and 30° after my original answer.
this creates 2 right-angled triangles.
the right angle is in both cases the angle where house meets the ground.
they also share one leg : the height of the house (10 m).
the second legs are the ground distances of the cars from the house.
the 2 Hypotenuses are the line of sight from the roof to the corresponding car.
remember, the sum of all angles in a triangle is always 180°.
again, we know one angle : the 90° angle.
but we also know a second angle based on the angles of depression (the "downward looking angles").
the triangle internal angle at the rooftop is the complementary angle (the difference to 90°) of the angle of depression.
so, this is 90-15 = 75° and 90-30 = 60°.
the angles at the cars on the ground are then
angle car 1 = 180 - 90 - 75 = 15°
angle car 2 = 180 - 90 - 60 = 30°
now, remember the trigonometric triangle inscribed in a circle.
imagine that the vertex at the car is the center of the corresponding circle around the trigonometric triangle.
the height of the house is then sine of the angle at the car multiplied by the Hypotenuse (= the line is sight from the rooftop to the car), which is the angle creating radius of the circle.
and the ground distance is the cosine of that same angle multiplied by the Hypotenuse.
so, we need to get the ratio of the height of the house / sin(car angle) to get the length of the Hypotenuse (line of sight). with that we can then calculate the ground distance as cosine of the angle multiplied by the same Hypotenuse.
for car 1 we have
10m/sin(15) = 38.63703305... m line of sight
that means ground distance of car 1 is
cos(15)×38.63703305... = 37.32050808... m
for car 2 we have
10m/sin(30) = 20 m line of sight
that means ground distance of car 2 is
cos(30)×20 = 17.32050808... m
since both cars are driving on the same side of the house in the same direction, the distance between both cars is purely the difference between their distances from the house :
37.32050808... - 17.32050808... = 20 m
the cars are 20 m apart.
and now for the angles of depression of 45° and 30° :
the triangle internal angle at the rooftop is the complementary angle (the difference to 90°) of the angle of depression.
so, this is 90-45 = 45° and 90-30 = 60°.
the angles at the cars on the ground are then
angle car 1 = 180 - 90 - 45 = 45°
angle car 2 = 180 - 90 - 60 = 30°
now, remember the trigonometric triangle inscribed in a circle.
imagine that the vertex at the car is the center of the corresponding circle around the trigonometric triangle.
the height of the house is then sine of the angle at the car multiplied by the Hypotenuse (= the line is sight from the rooftop to the car), which is the angle creating radius of the circle.
and the ground distance is the cosine of that same angle multiplied by the Hypotenuse.
so, we need to get the ratio of the height of the house / sin(car angle) to get the length of the Hypotenuse (line of sight). with that we can then calculate the ground distance as cosine of the angle multiplied by the same Hypotenuse.
for car 1 we have
10m/sin(45) = 14.14213562... m line of sight
that means ground distance of car 1 is
cos(45)×14.14213562... = 10 m
logically, as for 45° sine and cosine are equal.
for car 2 we have
10m/sin(30) = 20 m line of sight
that means ground distance of car 2 is
cos(30)×20 = 17.32050808... m
since both cars are driving on the same side of the house in the same direction, the distance between both cars is purely the difference between their distances from the house :
17.32050808... - 10 = 7.32050808... m
≈ 7.32 m
the cars are about 7.32 m apart.