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381.
Contrapositive of an Implication
$(p \implies q) \iff (\lnot q \implies \lnot p)$
Logic
382.
Equivalence as double implication
$(p \iff q) \iff \left[(p \implies q) \land (q \implies p)\right]$
Logic
383.
Equivalence is transitive
$(p \iff q) \land (q \iff r) \iff (p \iff r)$
Logic
384.
De Morgan's laws for conjunction
$\lnot ( p \land q) \iff \lnot p \lor \lnot q$
Logic
385.
De Morgan's laws for disjunction
$\lnot (p \lor q) \iff \lnot p \land \lnot q$
Logic
386.
Sum of opposite angles
$\alpha + \gamma = 180\degree$
Cyclic quadrilateral
387.
Sum of opposite angles
$\beta + \gamma = 180\degree$
Cyclic quadrilateral
388.
Ptolemy's theorem
$ac+bd=d_1d_2$
Cyclic quadrilateral
389.
Permeter
$P = a+ b+ c+ d$
Cyclic quadrilateral
390.
Circumscribed radius
$R=\dfrac{1}{4} \sqrt{ \dfrac{(ac+bd)(ad+bc)(ab+cd)}{(p-a)(p-b)(p-c)(p-d)} }, \text{ where } p = \dfrac{a+b+c+d}{2}$
Cyclic quadrilateral
391.
Area
$A = \dfrac{1}{2} d_1d_2 \sin \psi$
Cyclic quadrilateral
392.
Area
$A = \sqrt{(p-a)(p-b)(p-c)(p-d)}, \text{ where } p=\dfrac{a+b+c+d}{2}$
Cyclic quadrilateral
393.
$\sin^2\alpha = \dfrac{1-\cos2\alpha}{2}$
Power
Trigonometry
394.
$\sin^3\alpha = \dfrac{3\sin\alpha - \sin 3\alpha}{4}$
Power
Trigonometry
395.
$\sin^4\alpha = \dfrac{\cos4\alpha-4\cos2\alpha+3}{8}$
Power
Trigonometry
396.
$\sin^5\alpha = \dfrac{10\sin\alpha - 5\sin3\alpha + \sin 5\alpha}{16}$
Power
Trigonometry
397.
$\sin^6\alpha = \dfrac{10-15\cos2\alpha+6\cos4\alpha-\cos6\alpha}{32}$
Power
Trigonometry
398.
$\cos^2\alpha=\dfrac{1+\cos2\alpha}{2}$
Power
Trigonometry
399.
$\cos3\alpha=\dfrac{3\cos\alpha+\cos3\alpha}{4}$
Power
Trigonometry
400.
$\cos^4\alpha=\dfrac{\cos4\alpha+4\cos2\alpha+3}{8}$
Power
Trigonometry

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