Precalculus Trigonometry: Course Syllabus and Study Guide

Course Overview

This course, Pre-Calculus Trigonometry (MATH 1126), is designed to provide students with a comprehensive understanding of trigonometric concepts, their properties, and applications. The course is delivered online and covers essential topics in trigonometry, analytic geometry, and introductory topics in polar coordinates and the binomial theorem.

Course Structure and Weekly Topics

The course is organized into weekly modules, each focusing on specific sections from the textbook "Precalculus, 12th Edition by Sullivan." Below is a summary of the main topics and their sequence:

• Angles and Their Measure (Section 6.1)

• Trigonometric Functions: Unit Circle Approach (Section 6.2)

• Properties of Trigonometric Functions (Section 6.3)

• Graphs of Sine and Cosine Functions (Section 6.4)

• Phase Shift of Sine and Cosine Functions (Section 6.6)

• Graphs of Tangent, Cotangent, etc. (Section 6.5)

• Inverse Trigonometric Functions (Sections 7.1, 7.2)

• Trigonometric Equations and Identities (Sections 7.3, 7.4, 7.5, 7.6)

• Right Triangle Trigonometry (Section 8.1)

• Law of Sines and Law of Cosines (Sections 8.2, 8.3)

• Complex Numbers (A7)

• Polar Coordinates (Sections 9.1, 9.3)

• Binomial Theorem (Section 12.5)

Key Topics and Concepts

Angles and Their Measure

Angles are fundamental in trigonometry, measured in degrees or radians. Understanding how to convert between these units and interpret angles on the unit circle is essential.

• Degree Measure: A full circle is 360 degrees.

• Radian Measure: A full circle is $2\pi$ radians.

• Conversion Formula:

$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

• Example: Convert $90^\circ$ to radians: $90 \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.

The Unit Circle and Trigonometric Functions

The unit circle is a circle of radius 1 centered at the origin. It is used to define the six trigonometric functions for all real numbers.

• Sine and Cosine: For an angle $\theta$, the coordinates $(x, y)$ on the unit circle are $(\cos \theta, \sin \theta)$.

• Other Functions: $\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\csc \theta = \frac{1}{\sin \theta}$

• Example: At $\theta = \frac{\pi}{4}$, $\sin \theta = \cos \theta = \frac{\sqrt{2}}{2}$

Properties and Graphs of Trigonometric Functions

Trigonometric functions have specific properties such as period, amplitude, and phase shift. Their graphs are essential for understanding their behavior.

• Period: The period of $\sin$ and $\cos$ is $2\pi$.

• Amplitude: The maximum value from the midline, typically 1 for $\sin$ and $\cos$.

• Phase Shift: Horizontal shift of the graph.

• General Form: $y = A \sin(Bx + C) + D$

$\text{Period} = \frac{2\pi}{|B|}$

$\text{Phase Shift} = -\frac{C}{B}$

• Example: $y = 2 \sin(3x - \frac{\pi}{2}) + 1$ has amplitude 2, period $\frac{2\pi}{3}$, phase shift $\frac{\pi}{6}$, and vertical shift 1.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given trigonometric values. They are denoted as $\arcsin$, $\arccos$, and $\arctan$.

• Principal Values: Each inverse function has a restricted range to ensure it is a function.

• Example: $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$

Trigonometric Equations and Identities

Solving trigonometric equations and using identities is a core skill in trigonometry.

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$

• Sum and Difference Formulas:

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

• Double Angle Formulas:

$\sin 2\theta = 2 \sin \theta \cos \theta$

$\cos 2\theta = \cos^2 \theta - \sin^2 \theta$

• Example: Solve $2 \sin x = 1$ for $x$ in $[0, 2\pi)$: $\sin x = \frac{1}{2} \implies x = \frac{\pi}{6}, \frac{5\pi}{6}$

Right and Oblique Triangles

Trigonometry is used to solve triangles, both right and oblique (non-right) triangles.

• Right Triangle Trigonometry: Use definitions of sine, cosine, and tangent in terms of side lengths.

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

• Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$

• Example: Given $a = 7$, $b = 10$, $C = 45^\circ$, find $c$ using Law of Cosines.

Polar Coordinates and Complex Numbers

Polar coordinates represent points in the plane using a radius and angle. Complex numbers can be represented in polar form.

• Polar Coordinates: $(r, \theta)$ where $r$ is the distance from the origin and $\theta$ is the angle from the positive x-axis.

• Conversion: $x = r \cos \theta$, $y = r \sin \theta$

• Complex Numbers: $z = a + bi$ can be written as $z = r(\cos \theta + i \sin \theta)$

• Example: Convert $z = 1 + i$ to polar form: $r = \sqrt{2}$, $\theta = \frac{\pi}{4}$

Binomial Theorem

The Binomial Theorem provides a formula for expanding powers of binomials.

• Formula: $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

• Example: Expand $(x + 2)^3$ using the theorem.

Grading and Assessment

Component	Weight
Homework	25%
Tests (3 total)	50%
Final Exam	25%

Grading Scale:

Percentage	Grade
90% - 100%	A
80% - 89%	B
65% - 79%	C
55% - 64%	D
54% and below	F

Student Learning Outcomes

• Understand and apply angle measurement in degrees and radians.

• Use the unit circle to define and evaluate trigonometric functions.

• Graph trigonometric functions and analyze their properties.

• Solve trigonometric equations and use trigonometric identities.

• Apply trigonometry to right and oblique triangles.

• Work with polar coordinates and complex numbers.

• Use the Binomial Theorem for algebraic expansions.

Study and Success Tips

• Attend all lectures and participate actively.

• Review notes and textbook sections regularly.

• Complete all homework assignments and practice additional problems.

• Utilize office hours and academic support resources.

• Prepare for exams by reviewing examples, working practice problems, and using posted review materials.

Additional Information

• Calculator: Only TI-30X IIS or below is allowed.

• Homework is primarily completed on MyMathLab (MML).

• Academic integrity is strictly enforced; use of generative AI is not permitted.

• Accommodations are available for students with disabilities.

• Support services include tutoring, counseling, and accessibility services.