Double Angle Formulas/Cosine

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Theorem

$\cos 2 \theta = \cos^2 \theta - \sin^2 \theta$

where $\cos$ and $\sin$ denote cosine and sine respectively.


Corollary 1

$\cos 2 \theta = 2 \cos^2 \theta - 1$


Corollary 2

$\cos 2 \theta = 1 - 2 \sin^2 \theta$


Corollary 3

$\cos 2 \theta = \dfrac {1 - \tan^2 \theta} {1 + \tan^2 \theta}$


Corollary 4

$1 + \cos \theta = 2 \cos^2 \dfrac \theta 2$


Corollary 5

$1 - \cos \theta = 2 \sin^2 \dfrac \theta 2$


Proof 1

\(\ds \cos 2 \theta + i \sin 2 \theta\) \(=\) \(\ds \paren {\cos \theta + i \sin \theta}^2\) De Moivre's Formula
\(\ds \) \(=\) \(\ds \cos^2 \theta + i^2 \sin^2 \theta + 2 i \cos \theta \sin \theta\)
\(\ds \) \(=\) \(\ds \cos^2 \theta - \sin^2 \theta + 2 i \cos \theta \sin \theta\)
\(\ds \leadsto \ \ \) \(\ds \cos 2 \theta\) \(=\) \(\ds \cos^2 \theta - \sin^2 \theta\) equating real parts

$\blacksquare$


Proof 2

\(\ds \cos 2 \theta\) \(=\) \(\ds \map \cos {\theta + \theta}\)
\(\ds \) \(=\) \(\ds \cos \theta \cos \theta - \sin \theta \sin \theta\) Cosine of Sum
\(\ds \) \(=\) \(\ds \cos^2 \theta - \sin^2 \theta\)

$\blacksquare$


Proof 3

Starting from the right, we have:

\(\ds \cos^2 \theta - \sin^2\theta\) \(=\) \(\ds \paren {\frac 1 2 \paren {e^{i \theta} + e^{-i \theta} } }^2 - \paren {\frac 1 {2 i} \paren {e^{i \theta} - e^{-i \theta} } }^2\) Euler's Cosine Identity, Euler's Sine Identity
\(\ds \) \(=\) \(\ds \frac 1 4 \paren {e^{i \theta} + e^{-i \theta} }^2 + \frac 1 4 \paren {e^{i \theta} - e^{-i \theta} }^2\) $i$ is the imaginary unit
\(\ds \) \(=\) \(\ds \frac 1 4 \paren {e^{2 i \theta} + 2 + e^{-2 i \theta} + e^{2 i \theta} - 2 + e^{-2 i \theta} }\) Square of Sum, Square of Difference
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {e^{2 i \theta} + e^{-2 i \theta} }\) Simplifying
\(\ds \) \(=\) \(\ds \cos 2 \theta\) Euler's Cosine Identity

$\blacksquare$


Proof 4

Double angle sin.png

Consider an isosceles triangle $\triangle ABC$ with base $BC$, and apex $\angle BAC = 2 \alpha$.

Draw an angle bisector to $\angle BAC$ and name it $AH$.

$\angle BAH = \angle CAH = \alpha$

From Angle Bisector and Altitude Coincide iff Triangle is Isosceles:

$AH \perp BC$

From Law of Cosines:

\(\text {(1)}: \quad\) \(\ds CB^2\) \(=\) \(\ds AC^2 + AB^2 - 2 \cdot AB \cdot AC \cdot \cos 2 \alpha\)


From Pythagoras's Theorem:

\(\ds AC ^ 2\) \(=\) \(\ds CH^2 + AH^2\) in triangle $\triangle AHC$
\(\text {(2.1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds CH^2\) \(=\) \(\ds AC^2 - AH^2\)
\(\ds \) \(\) \(\ds \)
\(\ds AB ^ 2\) \(=\) \(\ds BH^2 + AH^2\) in triangle $\triangle AHB$
\(\text {(2.2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds BH^2\) \(=\) \(\ds BC^2 - AH^2\)


By definition of sine:

\(\text {(3.1)}: \quad\) \(\ds CH\) \(=\) \(\ds AC \sin \alpha\)
\(\text {(3.2)}: \quad\) \(\ds BH\) \(=\) \(\ds AB \sin \alpha\)


By definition of cosine:

$AH = AB \cos \alpha = AC \cos \alpha$


So:

\(\text {(4)}: \quad\) \(\ds AH^2\) \(=\) \(\ds AB \cdot AC \cdot \cos^2 \alpha\)
\(\ds \) \(\) \(\ds \)
\(\ds CH^2\) \(=\) \(\ds AC^2 - AH^2\) $(2.1)$
\(\text {(5.1)}: \quad\) \(\ds \) \(=\) \(\ds AC^2 - AB \cdot AC \cdot \cos^2 \alpha\) assigning $(4)$
\(\ds \) \(\) \(\ds \)
\(\ds BH^2\) \(=\) \(\ds AB^2 - AH^2\) $(2.2)$
\(\text {(5.2)}: \quad\) \(\ds \) \(=\) \(\ds AB^2 - AB \cdot AC \cdot \cos^2 \alpha\) assigning $(4)$


Now:

\(\ds CB^2\) \(=\) \(\ds (CH + BH)^2\)
\(\ds \) \(=\) \(\ds CH^2 + BH^2 + 2 \cdot CH \cdot BH\) Square of Sum
\(\ds \) \(=\) \(\ds AC^2 - AB \cdot AC \cdot \cos^2 \alpha + AB^2 - AB \cdot AC \cdot \cos^2 \alpha + 2 \cdot CH \cdot BH\) assigning $(5.1)$,$(5.2)$
\(\ds \) \(=\) \(\ds AC^2 + AB^2 - 2 \cdot AB \cdot AC \cdot \cos^2 \alpha + 2 \cdot CH \cdot BH\) simplifying
\(\ds \) \(=\) \(\ds AC^2 + AB^2 - 2 \cdot AB \cdot AC \cdot \cos^2 \alpha + 2 \cdot AB \cdot AC \cdot \sin^2 \alpha\) assigning $(3.1)$,$(3.2)$
\(\ds \) \(=\) \(\ds AC^2 + AB^2 - 2 \cdot AB \cdot AC \paren {\cos^2 \alpha - \sin^2 \alpha}\) simplifying
\(\ds \) \(=\) \(\ds AC^2 + AB^2 - 2 \cdot AB \cdot AC \cdot \cos 2 \alpha\) equating to $(1)$


Hence we get the equation:

\(\ds AC^2 + AB^2 - 2 \cdot AB \cdot AC \paren {\cos^2 \alpha - \sin^2 \alpha}\) \(=\) \(\ds AC^2 + AB^2 - 2 \cdot AB \cdot AC \cdot \cos 2 \alpha\)
\(\ds \leadsto \ \ \) \(\ds \cos^2 \alpha - \sin^2 \alpha\) \(=\) \(\ds \cos 2 \alpha\) simplifying

$\blacksquare$


Also known as

Corollary $1$ and Corollary $2$ are sometimes known as Carnot's Formulas, for Lazare Nicolas Marguerite Carnot.


Also see


Sources

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