BackPrecalculus Study Guide: Trigonometry, Graphs, and Exponential Models
Study Guide - Smart Notes
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Trigonometric Functions and Their Properties
Calculator Use and Evaluation of Trigonometric Functions
This section covers the evaluation of trigonometric functions using a calculator and understanding their properties for given angles.
Cotangent, Cosine, and Cosecant: Use a calculator to find values for cotangent, cosine, and cosecant for given arguments (in degrees or radians).
Example: Find , , , using calculator functions.
Finding All Trigonometric Functions from One Value
Given one trigonometric function value and quadrant information, you can determine the other five trigonometric functions.
Key Steps:
Identify the given function and its value (e.g., , in the third quadrant).
Use Pythagorean identities and quadrant signs to find the remaining functions: sine, cosine, cotangent, secant, and cosecant.
Example: If and is in the third quadrant, then and are both negative.
Trigonometric Functions from Coordinates
When the terminal side of an angle in standard position passes through a point , you can find all six trigonometric functions.
Key Formulas:
Example: For , .
Right Triangle Trigonometry
Given two side lengths of a right triangle, you can find the trigonometric functions of its angles.
Key Steps:
Label the sides as opposite, adjacent, and hypotenuse.
Use definitions:
Solving Trigonometric Equations
Solving for Angles
Trigonometric equations can be solved for specific intervals or sets.
Example: Solve for in or .
General Solution: ,
Graphing Trigonometric Functions
Graphing Cosine Functions
Understanding the properties of cosine functions is essential for graphing and analysis.
General Form:
Amplitude:
Period:
Phase Shift:
Example: For :
Amplitude: $2$
Period:
Phase Shift:
Graph Sketching
Sketch the graph using amplitude, period, and phase shift. Mark key points and axes.
Exponential Models: Newton's Law of Cooling and Radioactive Decay
Newton's Law of Cooling
Newton's Law of Cooling models the temperature change of an object in a surrounding environment.
Formula:
Variables:
: Temperature at time
: Surrounding temperature
: Initial temperature
: Cooling constant (negative value)
Example Table:
Time | Temperature |
|---|---|
First Data Point: | |
Second Data Point: |
Application: Use two data points to solve for and write the temperature formula. Predict temperature at minutes.
Radioactive Decay
Radioactive decay is modeled by an exponential function describing the quantity of an isotope over time.
Formula:
Variables:
: Quantity at time
: Initial quantity
: Decay constant (negative value)
Example: If mg at 8AM, mg at 11AM ( hours), solve for and write the decay formula.
Half-life: The time required for half the isotope to decay. Formula:
Summary Table: Exponential Models
Model | Formula | Key Parameters | Application |
|---|---|---|---|
Newton's Law of Cooling | , , | Temperature change over time | |
Radioactive Decay | , | Isotope quantity over time |
Additional info: The study notes expand on brief question prompts to provide full academic context, definitions, and formulas for Precalculus students. All equations are provided in LaTeX format for clarity.
