Converting Between Radians and Degrees

This page covers converting between radian and degree measures of angles.

Formulas are provided for converting from degrees to radians:

radians = $π/180$ * degrees

And from radians to degrees:

degrees = $180/π$ * radians

Example: 1 radian ≈ 57.2958°

The page also introduces trigonometric ratios for acute angles in a right triangle:

sine = opposite / hypotenuse cosine = adjacent / hypotenuse tangent = opposite / adjacent

A worked example demonstrates calculating these ratios for a 3-4-5 right triangle.

Highlight: Memorizing the ratios for common angles like 30°, 45° and 60° is very helpful for trigonometry problems.

Special Angle Trigonometric Ratios

This page focuses on the trigonometric ratios for the special angles of 45° and 60°.

For a 45-45-90 triangle: sin 45° = cos 45° = 1/√2 ≈ 0.7071 tan 45° = 1

For a 30-60-90 triangle: sin 60° = √3/2 ≈ 0.8660 cos 60° = 1/2 = 0.5 tan 60° = √3 ≈ 1.7321

Highlight: These special angle ratios appear frequently in trigonometry problems and should be memorized.

The fundamental trigonometric identity sin²θ + cos²θ = 1 is also introduced.

Definition: A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variables.

Trigonometric Relationships and the Unit Circle

This page explores relationships between trigonometric functions and introduces the unit circle.

Key concepts covered:

• The reciprocal relationships between trig functions $e.g. tan θ = sin θ / cos θ$
• Using the Pythagorean identity to find missing trig ratios
• Signs of trig functions in different quadrants of the unit circle

Example: If cos θ = 3/5, then sin θ = 4/5 and tan θ = 4/3

The unit circle is introduced as a powerful tool for understanding trigonometric functions for any angle.

Definition: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.

Highlight: On the unit circle, the x-coordinate of a point represents cos θ and the y-coordinate represents sin θ for angle θ.

Angle Relationships in Trigonometry

This page covers important relationships between angles and their trigonometric functions.

Key relationships explored:

• Supplementary angles $sum to 180°$
• Opposite angles
• Angles differing by 180°
• Complementary angles $sum to 90°$

Example: For supplementary angles: sin $180° - θ$ = sin θ cos $180° - θ$ = -cos θ tan $180° - θ$ = -tan θ

Highlight: These angle relationships are crucial for simplifying trigonometric expressions and solving complex problems.

Solving Triangles: The Law of Sines

This page introduces techniques for solving triangles that are not right triangles.

The law of sines is presented:

a / sin A = b / sin B = c / sin C

Where a, b, c are side lengths and A, B, C are opposite angles.

Definition: The law of sines relates the sides of a triangle to the sines of the opposite angles.

A worked example demonstrates using the law of sines to find an unknown side length given two angles and one side.

Highlight: The law of sines is particularly useful when you know two angles and any side of a triangle.

Solving Triangles: The Law of Cosines

This page continues with techniques for solving non-right triangles, focusing on the law of cosines.

The law of cosines is presented:

a² = b² + c² - 2bc cos A b² = a² + c² - 2ac cos B c² = a² + b² - 2ab cos C

Definition: The law of cosines relates the square of one side of a triangle to the other two sides and the cosine of the opposite angle.

A worked example shows how to use the law of cosines to find an unknown side length given two sides and the included angle.

Highlight: The law of cosines is a generalization of the Pythagorean theorem and is useful when you know two sides and the included angle of a triangle.

Applications of Trigonometry

This final page demonstrates practical applications of trigonometry to geometry problems.

An example problem involves finding the apothem and area of a regular pentagon given the side length.

The solution uses:

• The formula for the central angle of a regular polygon
• Trigonometric ratios in a right triangle
• Area formula for regular polygons

Vocabulary: The apothem of a regular polygon is the distance from the center to the midpoint of any side.

Highlight: Trigonometry has wide-ranging applications in geometry, physics, engineering, and many other fields.

Introduction to Trigonometry

This page introduces fundamental trigonometric concepts for 4th year ESO students.

The unit circle is presented, showing angles measured in degrees at key points $0°, 90°, 180°, 270°$. Two important angle measures are defined:

Definition: A degree is 1/90th of a right angle. It is denoted by the ° symbol.

Definition: A radian is the angle subtended at the center of a circle by an arc equal in length to the radius.

The relationship between radians and degrees is illustrated geometrically on the unit circle.

Highlight: Understanding radian measure is essential for advanced trigonometry and calculus.