Trigonometric Functions and Their Inverses

Inverse Trigonometric Functions

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

• arccos(x): The inverse of the cosine function, returns an angle whose cosine is x. Principal values: .

• arcsin(x): The inverse of the sine function, returns an angle whose sine is x. Principal values: .

• arctan(x): The inverse of the tangent function, returns an angle whose tangent is x. Principal values: .

• arccot(x): The inverse of the cotangent function, returns an angle whose cotangent is x.

• arcsec(x): The inverse of the secant function, returns an angle whose secant is x.

• arccsc(x): The inverse of the cosecant function, returns an angle whose cosecant is x.

Example: Find .

Evaluating Inverse Trigonometric Expressions

To evaluate expressions like , first find the angle whose secant is , then compute its sine.

• Step 1: Let so .

• Step 2:

• Step 3:

Trigonometric Identities and Simplification

Basic Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values in their domains.

• Pythagorean Identity:

• Quotient Identities: ,

• Reciprocal Identities: ,

Simplifying Trigonometric Expressions

Use identities to rewrite and simplify expressions.

• Example:

• Combine over a common denominator:

Trigonometric Functions: Odd, Even, or Neither

Classification of Functions

A function is even if , odd if , and neither if it satisfies neither property.

• Example:

• , so is odd.

Solving Right Triangles

Right Triangle Properties

In a right triangle, the relationships between the sides and angles are described by trigonometric ratios.

• Sine:

• Cosine:

• Tangent:

Example: Given , , find .

• Use Pythagorean Theorem:

Applications of Trigonometry

Angles of Elevation and Depression

Angles of elevation and depression are used to solve real-world problems involving heights and distances.

• Angle of Elevation: The angle formed by the horizontal and the line of sight looking up.

• Angle of Depression: The angle formed by the horizontal and the line of sight looking down.

• Example: From the top of a vertical cliff 80 meters above the ocean, the angle of depression to a marker on the surface is . Find the horizontal distance to the marker.

• Use

Factoring Trigonometric Expressions

Factoring Techniques

Factoring trigonometric expressions often involves recognizing patterns and using identities.

• Example:

• Factor out :

Table: Trigonometric Functions and Their Inverses

Additional info:

• Some problems require using a calculator to find decimal approximations for inverse trigonometric functions.

• Right triangle problems may involve using the Law of Sines or Law of Cosines for non-right triangles (not shown in this set).