Trigonometric Angles and Functions

Reference Angles and Radian Measure

Understanding angles in both degrees and radians is fundamental in trigonometry. Reference angles help simplify the evaluation of trigonometric functions for any angle.

• Reference Angle: The smallest angle between the terminal side of a given angle and the x-axis.

• Radian Measure: An alternative to degrees, where radians equals .

• Example: Find a positive angle less than that is coterminal with .

Trigonometric Functions and Their Properties

Trigonometric functions relate angles to ratios of sides in right triangles and are periodic in nature.

• Key Functions: Sine (), Cosine (), Tangent (), Cosecant (), Secant (), Cotangent ().

• Periodicity: Functions repeat every radians.

• Example: Use the unit circle and properties of trigonometric functions to find the value of .

Applications of Trigonometry

Right Triangle Applications

Trigonometric functions are used to solve problems involving right triangles, such as finding heights and distances.

• Example: A water wheel has a radius of 17 feet. The wheel is rotating at a constant rate. Find the linear speed of the water.

• Height Problems: If the height of a tree is found from a given angle, use , where is the distance and is the angle of elevation.

Trigonometric Identities and Equations

Fundamental Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable.

• Pythagorean Identity:

• Quotient Identity:

• Example: If and is in the right quadrant, find and .

Graphs of Trigonometric Functions

Amplitude, Period, and Phase Shift

The graphs of trigonometric functions can be transformed by changing their amplitude, period, and phase shift.

• Amplitude: The height from the center line to the peak.

• Period: The length of one complete cycle, for sine and cosine.

• Phase Shift: Horizontal shift of the graph.

• Example: Determine the amplitude, period, and phase shift of .

Graphing Trigonometric Functions

Graphing involves plotting the function over a specified interval and identifying key features.

• Example: Graph and over .

Inverse Trigonometric Functions

Definition and Evaluation

Inverse trigonometric functions allow us to find angles when given a trigonometric ratio.

• Notation: , ,

• Example: Evaluate .

Solving Trigonometric Equations

General Solutions

Trigonometric equations can have multiple solutions due to the periodic nature of the functions.

• Example: Solve for .

Applications: Law of Sines and Law of Cosines

Non-Right Triangle Solutions

The Law of Sines and Law of Cosines are used to solve triangles that are not right triangles.

• Law of Sines:

• Law of Cosines:

• Example: Given , , and , find the area of the triangle.

Additional info:

• Some questions involve graphing and interpreting trigonometric functions, which is essential for understanding their behavior and applications.

• Problems include both computational and conceptual aspects, such as using identities and solving equations.