Proving Common Trig Identities with Circle Theorems

I just re-read Trigonometric Delights by Eli Maor, and in the middle of his historical account of mathematics, he gives creative proofs for some common trigonometric identities based solely on Greek circle theorems.

Using the inscribed angle theorem, Ptolemy’s Theorem, and Thales’s Theorem, Maor is able to prove the law of sines, the angle addition and subtraction formulas for sine, the Pythagorean Theorem, and the Pythagorean identity of sine and cosine.  His proofs are outlined below.

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Consider the following triangle that is inscribed in a circle:

By the inscribed angle theorem, ∠AOB = 2∠ACB = 2γ.

Since OM bisects ∠AOB = 2γ, ∠MOB = γ.

Then from ΔOMB,

which rearranges to:

By a similar argument,

and

Therefore,

which proves the law of sines.

If 2r = 1, then a = sin α, b = sin β, and c = sin γ.  In other words, the side of an inscribed triangle in a circle with a unit diameter is equal to the sine of its opposite angle.

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Now consider the following diagram, where AC = 1:

Since 2r = 1, then as shown above BD is equal to the sine of its opposite angle, so BD = sin(α + β).

From ΔABC, cos α = AB/AC = AB/1 = AB and sin α = BC/AC = BC/1 = BC. 

Similarly from ΔADC, AD = cos β and CD = sin β.

But by Ptolemy’s Theorem, 

AC · BD = AB · CD + BC · DA

and after substituting the sine and cosine values, 

1 · sin(α + β) = cos α · sin β + sin α · cos β

which proves the angle addition formula for sine. 

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Similarly, consider the following diagram, where AD = 1:

Now BC is equal to the sine of its opposite angle, so BC = sin(α – β). 

From ΔABD, cos α = AB/AD = AB/1 = AB and sin α = BD/AD = BD/1 = BD. 

Similarly from ΔACD, AC = cos β and CD = sin β.

But by Ptolemy’s Theorem, 

AC · BD = AB · CD + BC · DA

and after substituting the sine and cosine values,

cos β · sin α = cos α · sin β + sin (α – β) · 1

which rearranges to:

sin(α – β) = sin α · cos β – cos α · sin β

which proves the angle subtraction formula for sine.

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Lastly, consider the following diagram of rectangle ABCD inscribed in a circle:

By the properties of a rectangle, AB = CD and AD = BC.

By Thales’s Theorem, AC and BD are diameters, so AC = BD.

By Ptolemy’s Theorem, 

AC · BD = AB · CD + BC · DA

and after substituting AB = CD, AD = BC, and AC = BD,

(AC)² = (AB)² + (BC)²

which proves the Pythagorean Theorem.

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Furthermore, if AC = 1, then cos α = AB/AC = AB/1 = AB and sin α = BC/AC = BC/1 = BC.

Substituting AC = 1, AB = cos α, and BC = sin α into the Pythagorean Theorem gives:

1 = cos²α + sin²α

which proves the Pythagorean identity of sine and cosine.

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Today’s math textbooks also have proofs for these trigonometric identities, but they are typically long and cumbersome.  Maor’s proofs are much more elegant, and also logically flow from a historical perspective.