Calculating the Angle Between the Sun's Diameter and Earth: Radians and Degrees

[XTEND]

The diameter of the sun is 1.39x10^9 m, and it is 1.5x10^11 m from the earth. When viewed from Earth what is the angle from one side of the sun to the other? Give answer is Radians and Degrees.


My problem is the qusetion also states do NOT use sine, cosine, or triangles to solve this.


My attempt was to think in spheres from one to the other drawing lines to connect the two, but then that results in a triangle.

HINT: the ANGLE is very small

TIA

 

[LowlyPion]

Welcome to PF.

Consider the small angle approximation.
http://en.wikipedia.org/wiki/Small-angle_approximation

 

[nlightner]

I would approach this problem by using basic geometry and trigonometry principles. However, since the question specifically states not to use sine, cosine, or triangles, I would approach it by using the concept of angular diameter.

The angular diameter is the angle subtended by an object at a given distance. In this case, the object is the sun and the distance is the distance between the sun and Earth. The formula for calculating angular diameter is:

θ = 2 arctan (d/2D)

Where θ is the angular diameter, d is the diameter of the object, and D is the distance between the object and the observer.

Plugging in the given values, we get:

θ = 2 arctan (1.39x10^9 m/2(1.5x10^11 m))

= 2 arctan (9.27x10^-3)

= 0.0185 radians

To convert this to degrees, we multiply by 180/π, which gives us:

θ = 0.0185 x (180/π)

= 1.06 degrees

Therefore, the angle between the sun's diameter and Earth when viewed from Earth is approximately 0.0185 radians or 1.06 degrees. This is a very small angle, as hinted in the question, which makes sense since the sun is much larger and farther away from Earth than any other object we observe in our daily lives.

I hope this helps and provides a satisfactory response. If you have any further questions or concerns, please do not hesitate to ask. Thank you.