"Published in Newark, California, USA"
Find the side opposite the given angle for a spherical triangle having
(a) b = 60°, c = 30°, A = 45°
(b) a = 45°, c = 30°, B = 120°
Solution:
A spherical triangle is a triangle whose sides are the edges of a sphere. It is not the same as a plane triangle because the sides of a spherical triangles are curve and not a straight line. A spherical triangle looks like this
 |
| Photo by Math Principles in Everyday Life |
As you notice that the measurements of the edges of a spherical triangle are expressed in degrees because the sides of a spherical triangle are the arcs of a sphere. The measurements of the arcs of a sphere or the edges of a spherical triangle are measured from the center of a sphere.
In this case, we will use the formulas that are completely different from the formulas of plane triangles. The formulas that we will use are the following:
The above formulas are called the Law of Cosines. You must remember or memorize all the above formulas because you will use these often in solving the sides and the angles of spherical triangles.
Now, let's go back to the given problem, if
(a) b = 60°, c = 30°, A = 45°
then we have to solve for the measurement of arc a. Use the first formula as follows
Substitute the values b, c, and A, we have
Let's have another one, if
(b) a = 45°, c = 30°, B = 120°
then we have to solve for the measurement of arc b. Use the second formula as follows
