Trigonometric Angles and Radian Measure

Degree and Radian Conversion

Understanding how to convert between degrees and radians is fundamental in trigonometry, as both units are used to measure angles.

• Degrees to Radians: Multiply the degree measure by .

• Radians to Degrees: Multiply the radian measure by .

• Example: Convert to radians: .

Evaluating Trigonometric Functions at Special Angles

Exact values of trigonometric functions at standard angles (such as , , , and their radian equivalents) are essential for solving many problems.

• Key Angles: , etc.

• Example: , .

Arc Length and Sector Area

Applications of Radian Measure

Radian measure is used to calculate arc lengths and areas of sectors in circles.

• Arc Length Formula: , where is arc length, is radius, and is in radians.

• Sector Area Formula: .

• Example: For cm and , .

The Unit Circle

Exact Values and Properties

The unit circle allows for the determination of trigonometric function values for any angle.

• Definition: A circle of radius 1 centered at the origin in the coordinate plane.

• Coordinates: For an angle , the coordinates on the unit circle are .

• Example: , .

Calculator Approximations

Decimal Values of Trigonometric Functions

Trigonometric functions can be evaluated using a calculator to obtain decimal approximations.

• Example: (rounded to five decimal places).

Linear and Angular Speed

Relating Rotational and Linear Motion

Linear speed and angular speed describe how fast a point moves along a circular path.

• Linear Speed (): , where is angular speed in radians per unit time.

• Angular Speed (): , where is in radians and is time.

• Example: If a drum of diameter 16.0 in rotates 52 times per minute, the angular speed is radians/minute.

Trigonometric Identities

Fundamental Identities

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

• Pythagorean Identities:

• Reciprocal Identities:

• Quotient Identities:

Verifying Trigonometric Identities

To verify an identity, manipulate one or both sides using algebraic and trigonometric properties until both sides are identical.

• Example: Verify .

Sum, Difference, Double, and Half-Angle Identities

Sum and Difference Identities

These identities allow the calculation of trigonometric functions for sums or differences of angles.

• Sine:

• Cosine:

• Tangent:

• Example:

Double-Angle Identities

Double-angle identities express trigonometric functions of in terms of .

Half-Angle Identities

Half-angle identities are used to find the sine, cosine, or tangent of half an angle.

Product-to-Sum and Sum-to-Product Identities

These identities convert products of sines and cosines into sums or differences, and vice versa.

• Product-to-Sum:

• Sum-to-Product:

Solving Trigonometric Equations

Trigonometric equations can be solved for all solutions in a given interval, often using identities and algebraic manipulation.

• Example: Solve for in .

Table: Common Trigonometric Values

The following table summarizes the exact values of sine and cosine for common angles:

Summary

• Trigonometry involves the study of angles, their measurement, and the relationships between the sides and angles of triangles.

• Key skills include converting between degrees and radians, evaluating trigonometric functions, applying identities, and solving equations.

• Applications include circular motion, arc length, sector area, and modeling periodic phenomena.