Precalculus Trigonometry: Syllabus, Key Concepts, and Reference Figures

Course Overview: Precalculus Trigonometry

Course Description

This course covers trigonometric functions, identities, equations, and their applications, including vectors, complex numbers, polar coordinates, and parametric equations. The curriculum is designed to prepare students for calculus and related fields in science and engineering.

• Prerequisites: Completion of Math 25 with a C or better.

• Emphasis: Graphing, solving equations, and applications in STEM and business.

Student Learning Outcomes

By the end of the course, students should be able to:

• Define and graph all trigonometric functions and their inverses; solve equations involving trigonometric functions both graphically and analytically.

• Use trigonometric identities, the Law of Sines, and the Law of Cosines to solve application problems.

• Define, graph, and solve problems involving vectors, complex numbers in trigonometric form, polar coordinates, and parametric equations.

Objectives

• Apply and graph trigonometric functions using right triangle and unit circle approaches.

• Analyze and graph inverse trigonometric functions.

• Solve trigonometric equations and verify identities.

• Graph transformations of trigonometric functions.

• Apply the Law of Sines and Law of Cosines to solve triangles.

• Use vectors and complex numbers in trigonometric form and perform operations.

• Work with polar coordinates and parametric equations.

Key Reference Figures

Special Right Triangles

Special right triangles are essential for understanding trigonometric ratios and the unit circle. The two most common types are the 45°-45°-90° triangle and the 30°-60°-90° triangle.

• 45°-45°-90° Triangle: Both legs are equal; hypotenuse is $\sqrt{2}$ times the length of a leg.

• 30°-60°-90° Triangle: The side opposite 30° is half the hypotenuse; the side opposite 60° is $\sqrt{3}/2$ times the hypotenuse.

Triangle Type	Angles	Side Lengths (relative to hypotenuse)
45°-45°-90°	45°, 45°, 90°	Legs: $1$, Hypotenuse:$\sqrt{2}$
30°-60°-90°	30°, 60°, 90°	Opposite 30°: $1$, Opposite 60°:$\sqrt{3}$, Hypotenuse: $2$

Know these triangles—they are foundational for trigonometric calculations and the unit circle.

The Unit Circle

The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. It is fundamental for defining the trigonometric functions for all real numbers.

• Coordinates: Each point on the unit circle corresponds to $(\cos \theta, \sin \theta)$ for an angle $\theta$ measured from the positive x-axis.

• Common Angles: $0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°$

• Radians: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{5\pi}{6}, \pi, \frac{7\pi}{6}, \frac{5\pi}{4}, \frac{4\pi}{3}, \frac{3\pi}{2}, \frac{5\pi}{3}, \frac{7\pi}{4}, \frac{11\pi}{6}, 2\pi$

Angle (Degrees)	Angle (Radians)	Coordinates $(x, y)$
0°	$0$	$(1, 0)$
90°	$\frac{\pi}{2}$	$(0, 1)$
180°	$\pi$	$(-1, 0)$
270°	$\frac{3\pi}{2}$	$(0, -1)$
360°	$2\pi$	$(1, 0)$
45°	$\frac{\pi}{4}$	$(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
30°	$\frac{\pi}{6}$	$(\frac{\sqrt{3}}{2}, \frac{1}{2})$
60°	$\frac{\pi}{3}$	$(\frac{1}{2}, \frac{\sqrt{3}}{2})$

Major Topics Covered (from Schedule)

• Trigonometric functions and their graphs

• Right triangle trigonometry

• Trigonometric identities and equations

• Law of Sines and Law of Cosines

• Vectors and complex numbers

• Polar coordinates and parametric equations

• Applications in geometry and real-world problems

Important Formulas

Trigonometric Ratios (Right Triangle)

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Law of Sines

• $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Law of Cosines

• $c^2 = a^2 + b^2 - 2ab \cos C$

Unit Circle Definitions

• $\sin \theta = y$

• $\cos \theta = x$

• $\tan \theta = \frac{y}{x}$ (where $x \neq 0$)

Example Application

Example: Find the sine and cosine of 45° using the unit circle.

• On the unit circle, 45° corresponds to $\frac{\pi}{4}$ radians.

• Coordinates: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

• $\sin 45° = \frac{\sqrt{2}}{2}$

• $\cos 45° = \frac{\sqrt{2}}{2}$

Additional info:

• Course includes use of MyMathLab for online assignments and quizzes.

• Homework is assigned but not collected; quizzes are graded.

• Attendance and participation are emphasized for success.

• Students are encouraged to use office hours and campus resources for support.