Trigonometric Functions and Angles in Standard Position

Definition and Sketching of Angles

Angles in standard position have their vertex at the origin and their initial side along the positive x-axis. The terminal side is determined by the angle's measure, which can be positive (counterclockwise) or negative (clockwise).

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

• Coordinates on Terminal Side: For a point (x, y) on the terminal side, the distance from the origin is .

• Trigonometric Functions:

• Coterminal Angles: Angles that share the same terminal side, differing by multiples of or radians.

Example: For in standard position, using a 30-60-90 triangle, if , , :

Special Right Triangles

Special right triangles are used to find exact trigonometric values for common angles.

• 30-60-90 Triangle: Side ratios are .

• 45-45-90 Triangle: Side ratios are .

Example: For , reference angle is , and in quadrant III, , , .

Trigonometric Functions for Arbitrary Points

Finding Trig Values from Coordinates

Given a point on the terminal side of an angle , calculate and then the six trigonometric functions.

• Example: For , .

Exact Values and Reference Angles

Solving for Angles Given Trig Values

To find all angles in or that satisfy a given trigonometric equation, use reference angles and quadrant information.

• Example:

• Reference angle:

• Since sine is negative in QIII and QIV: ,

Special Angles and Their Trig Values

Memorize the trigonometric values for , , , , and .

Trigonometric Identities

Reciprocal and Pythagorean Identities

These identities are fundamental for simplifying and solving trigonometric equations.

• Reciprocal Identities:

• Pythagorean Identities:

Applications of Trigonometry

Solving Right Triangles

Use trigonometric ratios to solve for unknown sides or angles in right triangles.

• Example: Given , , :

• Use , , or to find missing sides.

Word Problems Involving Trigonometry

Trigonometry is used to solve real-world problems involving heights, distances, and angles.

• Shadow Problem: If a building casts a 69 ft shadow at elevation, , so ft.

• Ladder Problem: For a 10 m wall and ladder 3.2 m from the wall, , .

• Navigation Problems: Use the Law of Cosines or Pythagorean Theorem to find distances between moving objects.

Unit Circle and Trigonometric Values

Unit Circle Basics

The unit circle is a circle of radius 1 centered at the origin. It is used to define trigonometric functions for all real numbers.

• Coordinates on the unit circle correspond to .

• Common angles are marked in both degrees and radians.

Finding Exact Values Using the Unit Circle

• Example: , , .

• Use reference angles and quadrant signs to determine values.

Graphing Trigonometric Functions

Identifying Key Features

Trigonometric functions can be transformed by changing amplitude, period, phase shift, and vertical translation.

• Amplitude: The height from the midline to the peak.

• Period: The length of one complete cycle.

• Phase Shift: Horizontal shift of the graph.

• Vertical Translation: Up or down shift of the graph.

Example: For :

• Amplitude: $5$

• Period:

• Phase Shift:

• Vertical Translation: $3$

Things to Memorize

• Ratios for 30-60-90 and 45-45-90 triangles

• Definitions of trigonometric functions in standard position

• Trigonometric identities

• Right triangle trigonometric functions

• Table of trigonometric values for special angles from to

Additional info: Some context and explanations have been expanded for clarity and completeness, including the use of reference angles, quadrant analysis, and applications in word problems.