Cosine of Sum/Proof 4
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Theorem
- $\map \cos {a + b} = \cos a \cos b - \sin a \sin b$
Proof
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$AB$, $AC$, $AE$, and $AD$ are radii of the circle centered at $A$.
Let $\angle BAC = a$ and $\angle DAC = \angle BAE = b$.
By Euclid's First Postulate, we can construct line segments $BD$ and $CE$.
By Euclid's second common notion, $\angle DAB = \angle CAE$.
Thus by Triangle Side-Angle-Side Equality, $\triangle DAB \cong \triangle CAE$.
Therefore, $DB = CE$.
We now assign Cartesian coordinates to the points $B$, $C$, $D$, and $E$:
| \(\ds B\) | \(=\) | \(\ds \tuple {1, 0}\) | ||||||||||||
| \(\ds C\) | \(=\) | \(\ds \tuple {\cos a, \sin a}\) | ||||||||||||
| \(\ds D\) | \(=\) | \(\ds \tuple {\map \cos {a + b}, \map \sin {a + b} }\) | ||||||||||||
| \(\ds E\) | \(=\) | \(\ds \tuple {\cos b, -\sin b}\) | Cosine Function is Even and Sine Function is Odd |
We use the definition of the distance function on the Euclidean space $\struct {\R^2, d}$ as defined by the Euclidean metric:
- $\forall x, y \in \R^2: \map d {x, y} = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$
where $x = \tuple {x_1, y_1}, y = \tuple {x_2, y_2}$.
Thus:
- $DB \cong CE \iff \map d {D, B} = \map d {C, E}$
So, plugging in the coordinates of $B, C, D, E$, we get:
| \(\ds \paren {\map \cos {a + b} } - 1)^2 + \map {\sin^2} {a + b}\) | \(=\) | \(\ds \paren {\cos a - \cos b}^2 + \paren {\sin a + \sin b}^2\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cos^2 \left({a + b}\right) + \sin^2 \left({a + b}\right)\) | \(\) | \(\ds \) | multiplying out left hand side | ||||||||||
| \(\ds {} - 2 \, \map \cos {a + b} + 1\) | \(=\) | \(\ds \paren {\cos a - \cos b}^2 + \paren {\sin a + \sin b}^2\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 1 - 2 \, \map \cos {a + b} + 1\) | \(=\) | \(\ds \paren {\cos a - \cos b}^2 + \paren {\sin a + \sin b}^2\) | Sum of Squares of Sine and Cosine | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 - 2 \, \map \cos {a + b}\) | \(=\) | \(\ds \cos^2 a - 2 \cos a \cos b + \cos^2 b\) | multiplying out right hand side | ||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \sin^2 a + 2 \sin a \sin b + \sin^2 b\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 - 2 \, \map \cos {a + b}\) | \(=\) | \(\ds 2 - 2 \cos a \cos b + 2 \sin a \sin b\) | Sum of Squares of Sine and Cosine | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \cos {a + b}\) | \(=\) | \(\ds \cos a \cos b - \sin a \sin b\) |
$\blacksquare$