Course Overview

This course, "Trigonometry with Analytic Geometry" (MATH 137), provides a comprehensive study of trigonometric concepts, analytic geometry, and their applications. The syllabus outlines the structure, major learning objectives, assessment methods, and key topical areas that align with a standard Precalculus curriculum.

Major Course Learning Objectives

• Angle Measurement: Convert between degrees and radians.

• Trigonometric Functions: Find values of trigonometric functions for any angle in degree or radian measure.

• Triangle Solutions: Solve right and oblique triangles using trigonometric methods, including the Law of Sines and Law of Cosines.

• Vectors: Use and apply properties of vectors in the plane.

• Graphing: Graph trigonometric functions and analyze their properties.

• Trigonometric Identities and Equations: Use identities to verify equivalencies and solve equations.

• Coordinate Systems: Convert between rectangular and polar coordinates.

• Complex Numbers: Apply properties of complex numbers in both rectangular and polar forms.

• Conic Sections: Determine features and graph circles, parabolas, ellipses, and hyperbolas from standard equations.

• Applications: Use the above principles to solve practical problems.

Key Topics and Subtopics

Angles and Trigonometric Functions

Understanding angles and their measurement is foundational to trigonometry. Trigonometric functions describe relationships between angles and side lengths in right triangles and extend to the unit circle.

• Degrees and Radians: Two common units for measuring angles. Conversion formula:

• Trigonometric Functions: Sine, cosine, tangent, cosecant, secant, and cotangent. Defined for all real numbers using the unit circle.

• Example: To convert 90° to radians: radians.

Right and Oblique Triangles

Solving triangles involves finding unknown sides or angles using trigonometric relationships.

• Right Triangle Trigonometry: Use definitions of sine, cosine, and tangent for acute angles.

• Law of Sines:

• Law of Cosines:

• Example: Given , , , find using the Law of Sines.

Vectors

Vectors are quantities with both magnitude and direction, essential in physics and engineering applications.

• Vector Representation: In component form:

• Magnitude:

• Direction:

• Example: A vector with components (3, 4) has magnitude .

Graphs of Trigonometric Functions

Graphing trigonometric functions helps visualize periodic behavior and transformations.

• Basic Graphs: Sine and cosine functions have amplitude, period, phase shift, and vertical shift.

• General Form:

• Period:

• Example: has amplitude 2 and period .

Trigonometric Identities and Equations

Identities are equations true for all values in the domain; solving trigonometric equations is essential for many applications.

• Pythagorean Identity:

• Angle Sum and Difference:

• Solving Equations: Use algebraic and trigonometric techniques to find all solutions in a given interval.

• Example: Solve for in : .

Rectangular and Polar Coordinates

Coordinate systems allow for different representations of points in the plane, useful for graphing and solving geometric problems.

• Rectangular (Cartesian) Coordinates:

• Polar Coordinates: , where is the distance from the origin and is the angle from the positive x-axis.

• Conversion Formulas: ,

• Example: Convert to rectangular: , .

Complex Numbers

Complex numbers extend the real numbers and are essential in advanced mathematics and engineering.

• Rectangular Form:

• Polar Form: or

• Multiplication in Polar Form:

• Example: in polar form: , , so .

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, parabolas, ellipses, and hyperbolas.

• Circle:

• Parabola:

• Ellipse:

• Hyperbola:

• Example: The equation represents a circle centered at the origin with radius 3.

Assessment and Grading Structure

The course uses a variety of assessment types to evaluate student understanding and mastery of the material.

Grading Scale

Additional Info

• Calculator Policy: Only scientific calculators (e.g., TI-30XIIS) are permitted; graphing calculators and internet-capable devices are not allowed during exams.

• Proctored Exams: Both midterm and final exams are proctored and require scheduling in advance.

• Formula Sheet: A formula sheet and a blank unit circle are provided during exams; memorization of the unit circle is required.

• Academic Integrity: Strict policies are in place regarding independent work and unauthorized assistance during assessments.

Summary Table: Major Topics Covered

Additional info: The syllabus structure and module schedule suggest a focus on chapters 7–11 of a standard Precalculus text, emphasizing trigonometry, analytic geometry, and related applications.