Comprehensive Trigonometry, Algebra, and Exponential Functions Review Guide

Trigonometric Simplification and Identities Cheat Sheet

Fundamental Identities

• Pythagorean Identities:

  • $\text{sin}^2(x) + \text{cos}^2(x) = 1$

  • $1 + \tan^2(x) = \text{sec}^2(x)$

  • $1 + \text{cot}^2(x) = \text{csc}^2(x)$

• Quotient Identities:

  • $\tan(x) = \frac{\text{sin}(x)}{\text{cos}(x)}$

  • $\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}$

• Reciprocal Identities:

  • $\text{sec}(x) = \frac{1}{\text{cos}(x)}$

  • $\text{csc}(x) = \frac{1}{\text{sin}(x)}$

  • $\text{cot}(x) = \frac{1}{\tan(x)}$

Simplification Techniques

1. Use Pythagorean identities to replace $1 - \text{cos}^2(x)$ or $\text{sec}^2(x) - 1$ with $\text{sin}^2(x)$ or $\tan^2(x)$ respectively.

2. Substitute expressions for $\tan(x)$, $\text{sec}(x)$, etc., when simplifying complex fractions.

3. Factor and reduce where applicable using identities.

Evaluating Trigonometric Values

• Quadrants:

  • I: \text{sin} > 0, \text{cos} > 0

  • II: \text{sin} > 0, \text{cos} < 0

  • III: \text{sin} < 0, \text{cos} < 0

  • IV: \text{sin} < 0, \text{cos} > 0

Solving Trigonometric Equations

1. Identify the function and its properties (periodicity, even/odd).

2. Use inverse functions to find general solutions.

3. Restrict solutions to the interval required (e.g., $0 \text{ to } 2\text{π}$).

Right Triangle Trigonometry

• Definitions:

  • Opposite side, adjacent side, and hypotenuse relate via sine, cosine, and tangent:

    • $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

    • $\text{sin}(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

    • $\text{cos}(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

Real-World Applications

• Use trigonometry for height/angle calculations with right triangle properties.

• Apply laws of cosine and sine for non-right triangles to solve for sides and angles.

Growth and Decay Functions

• Exponential Growth: $N(t) = N_0 e^{rt}$

• Decay Model: $N(t) = N_0 e^{-kt}$

• Understand logistic growth and carrying capacity in population models.