The Sine & Cosine “Square Root” Trick

Degrees	Radians	Sine	Cosine
0	0	\(\frac{\sqrt{0}}{2}=0\)	\(\frac{\sqrt{4}}{2}=1\)
30	\(\pi /6\)	\(\frac{\sqrt{1}}{2}=\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)
45	\(\pi /4\)	\(\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}\)	\(\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}\)
60	\(\pi /3\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{\sqrt{1}}{2}=\frac{1}{2}\)
90	\(\pi /2\)	\(\frac{\sqrt{4}}{2}=1\)	\(\frac{\sqrt{0}}{2}=0\)

Fundamental Constants

• \(\pi \approx 3.1415926535\)
  Mnemonic: “May(3) I(1) have(4) a(1) large(5) container(9) of(2) coffee(6) beans(5)?”
• \(e \approx 2.718281828459045\)
  Mnemonic: 2.7, followed by the year 1828 twice, then the angles of an isosceles right triangle (45, 90, 45).

Logarithm Cheat Sheet

• Product Rule: \(\log_b(xy) = \log_b x+\log_b y\)
• Quotient Rule: \(\log_b(x/y) = \log_b x-\log_b y\)
• Power Rule: \(\log_b(x^p) = p \log_b x\)
• Change of Base: \(\log_b x = \frac{\log_c x}{\log_c b}\)

The De Moivre Shortcut

To derive double angles without memorizing, use De Moivre’s Theorem for the case where \(n=2\):

\( (\cos \theta+i\sin \theta)^2=\cos(2\theta)+i\sin(2\theta)\)

The Expansion: Expanding the left side gives:

\(\cos^2 \theta+2i\sin \theta \cos \theta-\sin^2 \theta\)

The result: By Grouping the real and imaginary parts on both sides, we get the identities:

• Real (Cosine): \(\cos(2\theta)=\cos^2 \theta-\sin^2 \theta\)
• Imaginary (Sine): \(\sin(2\theta)=2 \sin\theta\cos \theta\)

The Empirical (Three-Sigma) Rule

In a normal distribution (Bell Curve), data is distributed around the mean (\(\mu\)) based on standard deviations (\(\sigma\)):

• \(\pm 1\sigma\): 68.2% of all outcomes.
• \(\pm 2\sigma\): 95.4% of all outcomes.
• \(\pm 3\sigma\): 99.7% of all outcomes.