Chapter 1: Angles and Radians
* Defines angles and radians as measures of rotation.
* Explains the relationship between degrees and radians.
* Introduces trigonometric functions: sine, cosine, and tangent.
* Example: A wheel rotates 90 degrees. Its angle in radians is 90 degrees * (π/180) = π/2 radians.
Chapter 2: Unit Circle and Trigonometric Functions
* Describes the unit circle as a circle with radius 1.
* Defines sine, cosine, and tangent as coordinates on the unit circle.
* Explores the periodic nature of trigonometric functions.
* Example: For an angle θ = π/3, the coordinates on the unit circle are (cos(π/3), sin(π/3)) = (1/2, √3/2).
Chapter 3: Inverse Trigonometric Functions
* Introduces the inverse trigonometric functions: arcsine, arccosine, and arctangent.
* Defines these functions as the angles that produce a given trigonometric value.
* Example: The arcsine of 1/2 is π/6, as sin(π/6) = 1/2.
Chapter 4: Trigonometric Identities
* Presents fundamental trigonometric identities, such as the Pythagorean identity and the double-angle formulas.
* Explores the use of identities to simplify trigonometric expressions.
* Example: The Pythagorean identity states that sin²(x) + cos²(x) = 1.
Chapter 5: Trigonometric Equations
* Solves trigonometric equations using algebraic and graphical methods.
* Introduces multiple solutions due to the periodic nature of trigonometric functions.
* Example: To solve sin(x) = 1, we have x = π/2 + 2nπ or x = 3π/2 + 2nπ, where n is an integer.
Chapter 6: Applications of Trigonometry
* Demonstrates the use of trigonometry in real-world applications, such as:
* Law of cosines to find unknown sides and angles in triangles
* Law of sines to solve problems involving oblique triangles
* Vector addition and subtraction
* Example: A surveyor uses trigonometry to find the height of a tower by measuring the distance to its base and the angle of elevation.
Chapter 7: Polar Coordinates
* Introduces polar coordinates as an alternative way to represent points in a plane.
* Defines the polar form of a complex number and the conversion between rectangular and polar coordinates.
* Example: The point (-1, 1) can be represented in polar coordinates as (√2, -π/4).
Chapter 8: Parametric Equations
* Describes parametric equations as functions that represent both coordinates of a point as functions of a parameter.
* Explores the use of parametric equations to generate graphs.
* Example: The parametric equations x = cos(t), y = sin(t) generate the unit circle.
