Page 2: Trigonometric Problem Solving

This page focuses on applying trigonometric functions in right triangle problems and solving various trigonometric equations and identities. It presents a series of exercises to reinforce understanding and develop problem-solving skills.

The problems cover a range of topics, including:

1. Calculating triangle dimensions using trigonometric ratios
2. Finding angles in right triangles given side lengths
3. Applying trigonometric functions in geometric contexts (e.g., circles, rectangles)
4. Solving trigonometric equations and identities

Example: Problem 5.1 asks to calculate the perimeter of a triangle ABC using given dimensions and trigonometric ratios, demonstrating practical application of sine and cosine functions.

Example: Problem 5.2 involves finding trigonometric ratios in a circle with a given diameter and chord, illustrating the connection between trigonometry and circle geometry.

The page also includes more advanced problems that require manipulation of trigonometric expressions and application of multiple trigonometric identities:

Example: Problem 5.3 asks to evaluate complex trigonometric expressions involving multiple functions and special angle values.

Highlight: Problems 5.5 and 5.8 focus on determining angles given specific trigonometric function values, reinforcing the relationship between different trigonometric ratios.

The exercises progress in difficulty, culminating in proofs and more complex geometric applications:

Example: Problem 5.11 asks to prove a trigonometric identity involving cotangent and cosine functions.

Example: Problem 5.12 requires proving several trigonometric identities, demonstrating the interconnectedness of various trigonometric functions and angles.

These problems provide comprehensive practice in applying trigonometric functions to solve geometric problems and manipulate trigonometric expressions.

Page 3: Trigonometric Functions of Any Planar Angle

This page introduces the concept of trigonometric functions for any planar angle, extending beyond acute angles in right triangles. It begins to explore how trigonometric functions are defined for angles in standard position on a coordinate plane.

The page starts with the definition of tangent for a general angle:

Definition: The tangent of an angle α is defined as a number that...

Unfortunately, the text is cut off, so the complete definition is not available. However, this introduction suggests that the document is expanding the concept of trigonometric functions to the unit circle and coordinate plane, which is a crucial step in developing a more comprehensive understanding of trigonometry.

This transition from right triangle trigonometry to general angle trigonometry is an important topic in advanced trigonometry, laying the foundation for topics such as:

• Trigonometric functions of obtuse angles
• Periodic properties of trigonometric functions
• Graphing trigonometric functions
• Trigonometric identities for any angle

Highlight: The extension of trigonometric functions to any planar angle is a key concept in trigonometry, enabling the study of periodic phenomena and more complex mathematical models.

While the full content is not visible, this page likely continues to explore how the definitions of sine, cosine, and other trigonometric functions are adapted for angles in standard position, using the coordinates of points on the terminal side of the angle.