Trigonometry question regarding the diameter of the sun?

Question

If the distance to the sun is approximately 93 million miles, and, from the earth, the sun subtends an angle of approximately 0.5 degrees, estimate the diameter of the sun to the neareset 10,000 miles.

Answer 1

Diameter is 2 times Radius
Radius = distance times tangent(angle)
2*93e6*tan(.5) = 1,623,200 miles (make sure the calculator is in degrees!)

Or the small angle approximation
Angle=Radius/distance ==> Radius=Angle*distance
.00872*93e6 = 811,600 miles (For this approximation, the angle MUST be in radians! check this out - the tan of .00872 = .00872)
Diameter = 2* Radius = 1,623,200 miles

This was fun!

Answer 2

The equations most people here are using are radius = distance * tan(angle/2) but this however is not totally correct.

At first glance this makes sense and it does work well enough when the angle is small, however it only thinks in two dimensions. One must remember that the object has depth.

The best way to illustrate this is to determine what angle the Earth subtends for an observer standing on it's surface. Common sense says 180 degrees, because we are standing on it. So, the formula angle = 2 * arctan(radius/distance) should give us 180 degrees, but it doesn't. It gives us 90 degrees, since radius and distance are practically equal (we're on the surface of the Earth, so our distance from its center is equal to it's radius, so 2 * arctan(1) = 90 degrees).

The most appropriate trig function in this case is the sine, not the tangent. The following illustration I created shows why:

http://public.clunet.edu/~sjfahmie/angles.gif

The top drawing shows what happens when you use the tangent. As you can see, the angle in question really isn't the angle subtended by the object itself, but it's radius. One must consider the depth of the object. The second illustration shows the more appropriate visualization, and shows why using the sine is more appropriate. Since 2 * arcsin(1) = 180 degrees, it will give the correct answer for standing on the Earth. For sufficiently small angles as I mentioned before, tan will give a similar answer to sin, but it's not technically correct.

Answer 3

It's a small angle, so you can use any small angle approximation:

Opp/Adj = tan(theta)

Answer 4

somebody call
Stephen Hawking.....