1.2. Function identities#
1.2.1. Trigonmetric functions#
(1.5)#\[\begin{align}
\sin \theta &= \frac{y}{r} & \csc \theta &= \frac{r}{x} = \frac{1}{\cos \theta} \\
\cos \theta &= \frac{x}{r} & \sec \theta &= \frac{r}{x} = \frac{1}{\cos \theta} \\
\tan \theta &= \frac{y}{x} = \frac{\sin \theta}{\cos \theta} &
\cot \theta &= \frac{x}{y} = \frac{\cos \theta}{\sin \theta}
\end{align}\]
(1.6)#\[\begin{align}
\sin^2 \theta + \cos^2 \theta = 1 \\
1 + \tan^2 \theta = \sec^2 \theta \\
1 + \cot^2 \theta = \csc^2 \theta
\end{align}\]
(1.7)#\[\begin{align}
\cos(A+B) = \cos A \cos B - \sin A \sin B \\
\sin(A+B) = \sin A \cos B - \cos A \sin B
\end{align}\]
(1.8)#\[\begin{align}
\cos 2 \theta = \cos^2 \theta - \sin^2 \theta \\
\sin 2 \theta = 2 \sin \theta \cos \theta
\end{align}\]
(1.9)#\[\begin{align}
\cos^2 \theta = \frac{1 + \cos 2 \theta}{2} \\
\sin^2 \theta = \frac{1 - \cos 2 \theta}{2}
\end{align}\]
(1.10)#\[\begin{align}
\sin(\theta + 2 \pi) = \sin \theta \\
\cos(\theta + 2 \pi) = \cos \theta
\end{align}\]
(1.11)#\[\begin{align}
\sin(- \theta) = - \sin \theta \\
\cos(- \theta) = cos \theta
\end{align}\]
1.2.2. Exponential functions#
(1.12)#\[\begin{align}
a^x a^y &= a^{x+y} \\
\frac{a^x}{a^y} &= a^{x-y} \\
(a^x)^y = (a^y)^x &= a^{xy} \\
a^x b^x &= (ab)^x \\
\frac{a^x}{b^x} &= \left (\frac{a}{b} \right)^x
\end{align}\]
1.2.3. Logarithmic functions#
Definition:
(1.13)#\[\begin{align}
y &= \log_{a}x \\
x &= a^y
\end{align}\]
Natural Log:
(1.14)#\[\begin{equation}
\ln x = \log_{e} x
\end{equation}\]
Common log:
(1.15)#\[\begin{equation}
\log x = \log_{10} x
\end{equation}\]
(1.16)#\[\begin{align}
\ln(bx) &= \ln(b) + \ln(x) \\
\ln\left(\frac{b}{x}\right) &= \ln(b) - \ln(x) \\
\ln(x^r) &= r \ln(x) \\
\log_{a}x &= \frac{\ln(x)}{\ln(a)} \\
\end{align}\]