The Law of Sines Made Clear

Adam Hrankowski

Once you see a triangle this way, you can never unsee it.

Do you remember the Law of Sines from high school trigonometry? You may have memorized it, but did you really get it? Did you feel it in your liver?

Try this.

Begin with a triangle. Label the vertices A, B and C. Label each side with the lower case letter corresponding to the opposite vertex.
Fig 1: A triangle.Image by Author

The Law of Sines says this:

Eq. 1: The Law of SinesImage by Author

We could prove this with a lot of convincing algebra, but what if we do this instead? Drop a line from Point C perpendicular to Side c. This divides our original triangle into two smaller triangles. The dividing line is common to both of the new triangles.

Label that line on one side in terms of Angle A; on the other side in terms of Angle B.

This is what you should now have.
Fig 2: Same triangle, dressed up a little.Image by Author

On the left, the dividing line has a length b sin A. On the right, it’s a sin B. Therefore:

Eq. 2Image by Author

But that is the Law of Sines. We could leave it at that, or convert it into the familiar ratios:

Eq. 3Image by Author

What about Angle C and Side c? We use a similar construction. In this case, the perpendicular will fall outside of the triangle. No worries.
Fig. 3Image by Author

This creates a new triangle outside of the original But what about Angle C? It is the exterior angle to the new triangle. We can label it C’. Our perpendicular is then a sin C’.
Fig. 4Image by Author

What good does that do us? Angles C and C’ are supplementary. (They add to 180⁰.) That’s good news. Here’s what that means for us:

Eq. 4Image by Author

This leaves us with the desired identity:

Eq. 5Image by Author

Conversion to ratios gives us this.

Eq. 6Image by Author

Combine with Equation 3. And there we have it. The Law of Sines.

Equation 7: Bob’s yer uncle.Image by Author

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