Trigonometric Functions and Their Properties

Degrees and Radians

Angles can be measured in either degrees or radians. Understanding how to convert between these units is essential for solving trigonometric problems.

• Degrees to Radians: Multiply the degree measure by .

• Radians to Degrees: Multiply the radian measure by .

• Example: Convert to radians: radians.

Arc Length Formula

The arc length of a circle subtended by a central angle (in radians) is given by:

• Formula:

• Where s is the arc length, r is the radius, and \theta is the angle in radians.

• Example: For a circle of radius 5 and angle , .

Right Triangle Trigonometry

Basic Trigonometric Ratios

In a right triangle, the six trigonometric functions are defined as ratios of the sides:

Given one trigonometric function and a right triangle, you can find the remaining functions using these ratios.

Trigonometric Values for Special Angles

Table of Values for , ,

The values of the six trigonometric functions for these special angles are fundamental for solving many problems.

Additional info: Values are exact and commonly used in trigonometric calculations.

Evaluating Trigonometric Expressions

Exact Values in Degrees and Radians

• Use the unit circle and special triangles to find exact values for trigonometric expressions.

• For example, .

Reference Angles and Coterminal Angles

• Reference Angle: The acute angle formed by the terminal side of the given angle and the x-axis.

• Coterminal Angles: Angles that share the same terminal side. Found by adding or subtracting multiples of or radians.

• Example: is coterminal with because .

Direction and Revolutions

• Angles measured counter-clockwise from the positive x-axis are positive; clockwise are negative.

• Angles exceeding one revolution ( or radians) can be reduced to a coterminal angle within one revolution.

Graphing Trigonometric Functions

Graphs of , , and

• Amplitude: The maximum value from the midline (for sine and cosine).

• Period: The length of one complete cycle.

  • For and : Period is .

  • For : Period is .

• Example: The graph of oscillates between and $1.

Inverse Trigonometric Functions

Evaluating Inverse Trig Expressions

• Inverse trig functions return the angle whose trigonometric function equals a given value.

• Example: .

Composite Functions Involving Inverse Trig Functions

• Evaluate expressions like by considering right triangle relationships.

• Example: can be found by constructing a right triangle with adjacent $3, and finding the opposite side using the Pythagorean theorem.

Signs of Trigonometric Functions in Quadrants

The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies:

• Quadrant I: All functions positive

• Quadrant II: Sine and cosecant positive

• Quadrant III: Tangent and cotangent positive

• Quadrant IV: Cosine and secant positive

Trigonometric Identities

Fundamental Identities

• Pythagorean Identities:

• Reciprocal Identities:

• Quotient Identities:

Using Identities to Find Exact Values

• Apply identities to simplify and evaluate trigonometric expressions.

• Example: If and is in Quadrant II, find using .

Additional info: Mastery of these identities is essential for solving trigonometric equations and simplifying expressions.