Sexagesimal System and Angle Measurements

This section explains the sexagesimal system for measuring angles and how to convert between different angle representations.

Key concepts:

• Incomplete and complete units in the sexagesimal system
• Converting between degrees, minutes, and seconds
• Adding and subtracting angles in sexagesimal form
• Handling angles greater than 360°

Example: To convert 32.75° to the sexagesimal system: 32° + (0.75 × 60') = 32° 45'

Definition: Sexagesimal system - An angle measurement system based on 60, using degrees, minutes, and seconds.

The page also demonstrates how to perform arithmetic operations with complex angle measurements and deal with angles exceeding 360°.

International System and Trigonometric Ratios

This section covers the conversion between degrees and radians, as well as the basic trigonometric ratios.

Key topics:

• Converting between degrees and radians
• Defining sine, cosine, tangent and their reciprocals
• Trigonometric ratios in different quadrants

Vocabulary:

• Radian - The angle subtended at the center of a circle by an arc equal in length to the radius
• Cosecant (cosec) - The reciprocal of sine
• Secant (sec) - The reciprocal of cosine
• Cotangent (cotg) - The reciprocal of tangent

Highlight: The conversion factor between degrees and radians is π/180°.

The page also includes a diagram showing how trigonometric ratios relate to the sides of a right triangle in each quadrant.

Fundamental Trigonometric Relationships

This section presents the core trigonometric identities and demonstrates their proofs.

Key relationships covered:

1. sin^2 x + cos^2 x = 1
2. tan x = sin x / cos x
3. 1 + tan^2 x = 1 / cos^2 x

Definition: Trigonometric identity - An equation involving trigonometric functions that is true for all values of the variables.

Example: Proof of sin^2 x + cos^2 x = 1 using the Pythagorean theorem in a right triangle.

The page also includes a table of trigonometric ratios for common angles (0°, 30°, 45°, 60°, 90°), which is essential for solving many trigonometry problems.

Trigonometric Ratios for Any Angle

This section extends trigonometric concepts to angles in all quadrants and provides practice exercises.

Key topics:

• Expressing angles in radians
• Determining signs of trigonometric functions in different quadrants
• Solving right triangle problems with various given information

Example: Express 240° in radians: 240° × $π rad / 180°$ = 4π/3 rad

Highlight: The signs of trigonometric functions change depending on the quadrant of the angle.

Practice exercises include:

• Converting between degrees and radians
• Finding missing sides in right triangles given an angle and one side
• Calculating unknown angles in right triangles

Trigonometry Problem Solving

This final section provides more complex problem-solving examples involving right triangles.

Key problem types:

• Finding the hypotenuse given an angle and opposite side
• Calculating cathetuses (legs) given the hypotenuse and an angle
• Determining missing sides and angles in right triangles

Example: In a right triangle with hypotenuse 26 cm and an angle of 66°, the opposite cathetus is calculated as 26 × sin(66°) ≈ 23.74 cm.

Vocabulary: Cathetus (plural: cathetuses) - A leg of a right triangle; the side adjacent to the right angle

The problems demonstrate how to apply trigonometric ratios and the Pythagorean theorem to solve for unknown elements in right triangles, reinforcing the practical application of the concepts covered throughout the document.

Page 6: General Angle Trigonometry

Focuses on trigonometric ratios for any angle and includes practical activities.

Example: Converting angles between degrees and radians using π.

Highlight: Signs of trigonometric functions in different quadrants are explained in detail.

Page 7: Problem Solving with Right Triangles

Contains practical exercises involving right triangle calculations.

Example: Finding the sides of a right triangle given an angle of 66° and hypotenuse of 26 cm.

Vocabulary: Terms related to triangle geometry and trigonometric calculations are introduced.

Trigonometry Fundamentals and Problem Solving

This section introduces core trigonometric concepts and problem-solving techniques for angles in different quadrants.

Key topics covered:

• Calculating sine and cosine values given other trigonometric ratios
• Solving for unknown sides and angles in right triangles
• Using trigonometric identities

Example: For an angle in the first quadrant with cosine 3/4, the sine is calculated as √$1 - cos^2$ = √$1 - (3/4)^2$ = √$7/16$ = √7/4.

Highlight: The fundamental trigonometric identity sin^2 x + cos^2 x = 1 is used frequently to solve problems.

Vocabulary: Tangent (tg) - The ratio of sine to cosine for an angle