Interactive Law of Sines
The law of sines is simple, easy to understand and to remember.
We know every triangle has 3 angles (hence tri-angle).
Let's call our 3 angles A, B, and C.
We also know every triangle has 3 side lengths.
Let's call our side length opposite of 'A' a, the side length opposite of 'B' b, and the side length opposite 'C' c.
The law of sines tells us that :
a/sin(A) = b/sin(B) = c/sin(C)
Why Does It Work
We remember that triangles are all about ratios.
We remember that the sine function takes an angle and returns us the ratio of the opposite over the hypotenuse.
But wait... doesn't that only work on right angles?
Correct.
How then does the law of sines work on all triangles?
Easy.
Any triangle can be divided into 2 right triangles.
As we can see above with the white dividing line (when we grab one of the triangle points) we divide our triangle into 2 right triangles and viola, we can now use sine functions.
Now we can do some simple algebra to see what is going on.
Where h is the length of the opposite side (the white line above).
sin(A) = h/b
so
b * sin(A) = h
likewise
sin(B) = h/a
so
a * sin(B) = h
The Ratios Relate to the Circumcircle
Drawing a circle that touches all of our triangle's points (as we are doing above) we find that each ratio is equal to the diameter of said circle.
a/sin(A) =
b/sin(B) =
c/sin(C) =
2 * radius-of-circle
Why Could The Law of Sines Be Useful
We can use the law of sines to find missing information of a given triangle.
If we are given an angle and it's opposite side length, then given one of the other angles we can find their corresponding opposite side length,
or given one of the other side lengths we can find their corresponding angle.
Also, as mentioned above, we can find the radius of a circumcircle.
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