• Versine ($\text{versin } \theta$): $1 - \cos \theta$. Historically used to measure the deviation of a curve from a straight line.
• Coversine ($\text{coversin } \theta$): $1 - \sin \theta$.
• Haversine ($\text{haversin } \theta$): $\frac{1 - \cos \theta}{2}$. Extremely important in navigation for calculating distances on a sphere (the Haversine formula).
• Exsecant ($\text{exsec } \theta$): $\sec \theta - 1$. Used extensively in railway engineering and surveying to calculate the external distance of a circular curve.
• Excosecant ($\text{excsc } \theta$): $\csc \theta - 1$.

• Sine ($\sin \theta$): $\displaystyle \frac{\text{Opposite}}{\text{Hypotenuse}}$ (SOH)
• Cosine ($\cos \theta$): $\displaystyle \frac{\text{Adjacent}}{\text{Hypotenuse}}$ (CAH)
• Tangent ($\tan \theta$): $\displaystyle \frac{\text{Opposite}}{\text{Adjacent}}$ (TOA)

• Cosecant ($\csc \theta$): $\displaystyle \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$
• Secant ($\sec \theta$): $\displaystyle \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$
• Cotangent ($\cot \theta$): $\displaystyle \frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}}$

• $\cos \theta = x$
• $\sin \theta = y$
• $\tan \theta = \frac{y}{x} \quad (x \neq 0)$

• $\sec \theta = \frac{1}{x} \quad (x \neq 0)$
• $\csc \theta = \frac{1}{y} \quad (y \neq 0)$
• $\cot \theta = \frac{x}{y} \quad (y \neq 0)$

• Tangent ($\tan \theta$): The length of the line segment that is strictly tangent to the circle at $(1, 0)$ extending up to the secant line.
• Secant ($\sec \theta$): The length of the hypotenuse from the origin intersecting the circle out to the tangent line at $x=1$.
• Cotangent ($\cot \theta$): The length of the horizontal tangent line at $(0, 1)$ extending to the secant line.
• Cosecant ($\csc \theta$): The length of the secant line from the origin intersecting the circle out to the horizontal tangent line at $y=1$.

• Even Functions: Symmetric about the y-axis. $f(-x) = f(x)$.
  • Cosine: $\cos(-\theta) = \cos(\theta)$
  • Secant: $\sec(-\theta) = \sec(\theta)$
• Odd Functions: Symmetric about the origin. $f(-x) = -f(x)$.
  • Sine: $\sin(-\theta) = -\sin(\theta)$
  • Tangent: $\tan(-\theta) = -\tan(\theta)$
  • Cosecant: $\csc(-\theta) = -\csc(\theta)$
  • Cotangent: $\cot(-\theta) = -\cot(\theta)$

• $\sin(x + 2\pi) = \sin(x)$
• $\cos(x + 2\pi) = \cos(x)$
• $\tan(x + \pi) = \tan(x)$

• Sine ($\sin$): Domain: $(-\infty, \infty)$. Range: $[-1, 1]$. Period: $2\pi$.
• Cosine ($\cos$): Domain: $(-\infty, \infty)$. Range: $[-1, 1]$. Period: $2\pi$.
• Tangent ($\tan$): Domain: $x \neq \frac{\pi}{2} + n\pi$. Range: $(-\infty, \infty)$. Period: $\pi$. Asymptotes at $x = \frac{\pi}{2} + n\pi$.
• Cosecant ($\csc$): Domain: $x \neq n\pi$. Range: $(-\infty, -1] \cup [1, \infty)$. Period: $2\pi$. Asymptotes at $x = n\pi$.
• Secant ($\sec$): Domain: $x \neq \frac{\pi}{2} + n\pi$. Range: $(-\infty, -1] \cup [1, \infty)$. Period: $2\pi$. Asymptotes at $x = \frac{\pi}{2} + n\pi$.
• Cotangent ($\cot$): Domain: $x \neq n\pi$. Range: $(-\infty, \infty)$. Period: $\pi$. Asymptotes at $x = n\pi$.

• Amplitude ($|A|$): Represents the peak deviation from the center line. It is half the distance between the maximum and minimum values. (Note: $\tan$, $\cot$, $\sec$, and $\csc$ do not have a defined amplitude, but $A$ affects their vertical stretch).
• Period ($P$): The horizontal length of one complete cycle.
  • For $\sin$, $\cos$, $\sec$, $\csc$: $P = \frac{2\pi}{|B|}$
  • For $\tan$, $\cot$: $P = \frac{\pi}{|B|}$
• Phase Shift ($C$): The horizontal shift. If $C \gt 0$, the graph shifts right; if $C \lt 0$, it shifts left. (Be careful with the negative sign in the formula).
• Vertical Shift ($D$): The vertical translation. The line $y = D$ becomes the new midline of the wave.

Important: In these formulas, the angle $x$ must be measured in radians.

• $\sin \theta \approx \theta$
• $\tan \theta \approx \theta$
• $\cos \theta \approx 1 - \frac{\theta^2}{2} \approx 1$

• Sine and Cosine: The domain is all real numbers $\mathbb{R}$. Because these functions map coordinates on the unit circle, their range is strictly $[-1, 1]$.
• Tangent and Secant: These functions involve division by $\cos \theta$. Therefore, their domain excludes all points where $\cos \theta = 0$ (i.e., odd multiples of $\frac{\pi}{2}$).
• Cotangent and Cosecant: These involve division by $\sin \theta$. Their domain excludes all points where $\sin \theta = 0$ (i.e., integer multiples of $\pi$).