Trigonometric Functions and Their Properties

Trig Ratios and Basic Equations

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides. These ratios are foundational for solving equations and modeling periodic phenomena.

• Sine (sin):

• Cosine (cos):

• Tangent (tan):

• Solving Basic Trig Equations: To solve , , or , use inverse functions and consider all possible solutions within the specified interval.

• Example: Solve for in .

  • Principal solution:

  • Other solution:

Graphing Trigonometric Functions

Attributes of Sine and Cosine Graphs

Understanding the attributes of trigonometric graphs is essential for modeling and interpreting periodic behavior.

• Amplitude: The maximum displacement from the midline. For , amplitude is .

• Period: The length of one complete cycle. For or , period is .

• Phase Shift: Horizontal shift, given by in .

• Vertical Shift: The value shifts the graph up or down.

• Example: For :

  • Amplitude: $3$

  • Period:

  • Phase Shift: to the right

  • Vertical Shift: $1$ unit up

Inverse Trigonometric Functions

Definitions, Domains, and Ranges

Inverse trigonometric functions allow us to find angles given a trigonometric ratio. Their domains and ranges are restricted to ensure they are functions.

• Arcsin (Inverse Sine):

  • Domain:

  • Range:

• Arccos (Inverse Cosine):

  • Domain:

  • Range:

• Arctan (Inverse Tangent):

  • Domain:

  • Range:

• Visualizing on the Unit Circle: The range of each inverse function corresponds to the principal values on the unit circle.

• Example:

Composition of Trigonometric and Inverse Functions

Algebraic Expressions and Simplification

Compositions such as can often be rewritten as algebraic expressions using right triangle relationships.

• Method: Let , so . Draw a right triangle with adjacent side and hypotenuse $1$.

• Find:

• Example:

Solving Trigonometric Equations

General and Interval Solutions

Trigonometric equations may have infinitely many solutions. Solutions are often found both algebraically and graphically.

• All Solutions: For , or , where is any integer.

• Interval Solutions: Restrict so that lies within the given interval (e.g., ).

• Graphical Solution: Plot and ; intersection points are solutions.

• Example: Solve for in .

Sinusoidal Equations and Modeling

Applications and Construction of Sinusoidal Functions

Sinusoidal functions model periodic phenomena such as sound waves, tides, and simple harmonic motion.

• General Form: or

• Given a Graph: Identify amplitude, period, phase shift, and vertical shift to write the equation.

• Simple Harmonic Motion: The position of an object in simple harmonic motion can be modeled as , where is angular frequency and is phase.

• Example: A mass oscillates with amplitude $5 s. The position is .

• Solving Problems: Use the model to find position at a given time or time for a given position.

Summary Table: Inverse Trigonometric Functions

Additional info:

• Content inferred and expanded based on standard precalculus curriculum for trigonometric functions, their graphs, and applications.

• Graphing attributes and modeling applications are essential for interpreting and constructing sinusoidal equations.