Trigonometric Functions and Their Properties

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given the value of a trigonometric function. The principal values are typically restricted to ensure the function is one-to-one.

• arcsin(x): The angle whose sine is x, with range .

• arccos(x): The angle whose cosine is x, with range .

• arctan(x): The angle whose tangent is x, with range .

Example: because .

Evaluating Trigonometric Expressions

To evaluate expressions such as , use right triangle relationships or the Pythagorean identity.

• Let . Then .

• Find using .

Example: (sign depends on quadrant).

Exact Values in Degrees

Some trigonometric values are commonly memorized for special angles (e.g., , , ).

Trigonometric Identities and Simplification

Basic Trigonometric Identities

• Pythagorean Identity:

• Quotient Identities: ,

• Reciprocal Identities: ,

Using Identities to Find Other Trig Functions

If one trigonometric function value is known and the quadrant is specified, use identities to find the remaining functions.

• Given , (sign depends on quadrant).

Simplifying and Proving Identities

To prove an identity, manipulate one or both sides using algebraic and trigonometric identities until both sides are equal.

• Example: Prove (requires algebraic manipulation and use of identities).

Even and Odd Functions

Definitions

• Even Function: for all in the domain.

• Odd Function: for all in the domain.

Examples: is even, is odd.

Factoring Trigonometric Expressions

Factoring is used to simplify expressions or solve equations. Look for common factors or use identities to rewrite expressions.

• Example:

Solving Right Triangles

Right Triangle Relationships

• Use the Pythagorean theorem:

• Trigonometric ratios: , ,

Example: Given , , find using .

Applications of Trigonometry

Angles of Elevation and Depression

These concepts are used to solve real-world problems involving heights and distances.

• Angle of Elevation: The angle above horizontal from the observer's eye to an object.

• Angle of Depression: The angle below horizontal from the observer's eye to an object.

Example: If a 150-foot tall building casts a shadow of 200 feet, the angle of elevation is found using .

Degree and Radian Measure

Conversion Between Degrees and Radians

• To convert degrees to radians:

• To convert radians to degrees:

Example: radians.

Summary Table: Common Trigonometric Values

Additional info: This table summarizes the most commonly used trigonometric values for special angles, which are essential for solving many precalculus problems.