Trigonometry: Core Concepts, Functions, and Applications – Study Notes

Course Overview

Introduction to Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right triangles. It is foundational for advanced mathematics, physics, engineering, and many applied sciences.

• Prerequisite: Completion of intermediate algebra or equivalent placement.

• Textbook: Trigonometry, 12th Edition by Lial, Hornsby, Schneider, & Daniels.

• Learning Outcomes:

  • Solve trigonometric equations and identities.

  • Graph trigonometric functions and analyze their properties.

  • Apply trigonometric relationships to solve real-world problems.

Fundamental Trigonometric Concepts

Angle Relationships and Similar Triangles

Understanding angles and their relationships is essential in trigonometry. Similar triangles have equal corresponding angles and proportional sides.

• Angle: Formed by two rays with a common endpoint.

• Degrees and Radians: Two units for measuring angles. $360^{\circ} = 2\pi$ radians.

• Similar Triangles: Triangles with the same shape but possibly different sizes.

• Application: Used in indirect measurement and geometric proofs.

Trigonometric Functions of Acute Angles

Trigonometric functions relate the angles of a triangle to the ratios of its sides. For acute angles in right triangles:

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

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

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

• Other Functions: Cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent, respectively.

Applications of Right Triangles

Trigonometry is used to solve problems involving right triangles, such as finding unknown sides or angles.

• Pythagorean Theorem: $a^2 + b^2 = c^2$

• Solving for Sides/Angles: Use trigonometric ratios and inverse functions.

• Example: Finding the height of a building using angle of elevation and distance from the base.

Trigonometric Functions and Their Graphs

Graphs of Sine, Cosine, and Tangent Functions

Graphing trigonometric functions helps visualize their periodic nature and key properties.

• Periodicity: Sine and cosine have a period of $2\pi$; tangent has a period of $\pi$.

• Amplitude: The maximum value from the midline (for sine and cosine).

• Phase Shift: Horizontal shift of the graph.

• Vertical Shift: Upward or downward movement of the graph.

• Example: $y = 2\sin(x - \frac{\pi}{4}) + 1$

Transformations of Trigonometric Graphs

Transformations include translations, stretches, and reflections.

• Translation: Shifting the graph horizontally or vertically.

• Stretch/Compression: Changing the amplitude or period.

• Reflection: Flipping the graph over the x-axis or y-axis.

Trigonometric Identities and Equations

Fundamental Trigonometric Identities

Identities are equations that are true for all values of the variable where both sides are defined.

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

• Quotient Identities: $\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$

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

Sum and Difference Identities

These identities allow the calculation of trigonometric functions for sums or differences of angles.

• Sine: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

• Cosine: $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

• Tangent: $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Double Angle and Half Angle Identities

These identities simplify expressions involving double or half angles.

• Double Angle: $\sin 2\theta = 2\sin \theta \cos \theta$

• Half Angle: $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

Inverse Trigonometric Functions

Definition and Properties

Inverse trigonometric functions allow the determination of angles from known ratios.

• Notation: $\sin^{-1} x$, $\cos^{-1} x$, $\tan^{-1} x$

• Domain and Range: Each inverse function has a restricted domain and range to ensure it is a function.

• Example: $\theta = \sin^{-1}(0.5)$

Applications of Trigonometry

Law of Sines and Law of Cosines

These laws are used to solve non-right triangles.

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

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

• Ambiguous Case: Occurs in the Law of Sines when given two sides and a non-included angle (SSA).

Area of a Triangle and Sector

Trigonometry provides formulas for calculating areas using side lengths and angles.

• Area of Triangle: $\text{Area} = \frac{1}{2}ab\sin C$

• Area of Sector: $\text{Area} = \frac{1}{2}r^2\theta$ (where $\theta$ is in radians)

Advanced Topics

Polar Coordinates and Graphs

Polar coordinates represent points in the plane using a radius and angle.

• Polar Form: $(r, \theta)$

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

• Graphing: Common polar graphs include circles, spirals, and roses.

DeMoivre's Theorem

DeMoivre's Theorem is used to raise complex numbers in polar form to powers.

• Theorem: $(r(\cos \theta + i\sin \theta))^n = r^n(\cos n\theta + i\sin n\theta)$

• Application: Simplifies computation of powers and roots of complex numbers.

Parametric Equations

Parametric equations express coordinates as functions of a parameter, often time.

• Form: $x = f(t)$, $y = g(t)$

• Application: Used to describe motion and curves in the plane.

Grading and Assessment

Grading Criteria

Course grades are based on a combination of homework, quizzes, exams, and class activities.

Category	Percentage
Homework	10%
Quizzes	10%
Exams	45%
Class Activities/Worksheets	25%
Final Exam	10%

Grading Scale

Percentage	Letter Grade
90-100%	A
80-89.9%	B
70-79.9%	C
60-69.9%	D
0-59.9%	F

Course Policies and Resources

Attendance and Participation

Regular attendance and active participation are required for success in trigonometry.

• Excessive absences may result in being dropped from the course.

• Electronic devices are not permitted during class unless authorized.

• Respectful behavior is expected at all times.

Academic Support

Students are encouraged to seek help during office hours or at the Math Center. Additional resources are available online.

Trigonometry Course Schedule & Topics

Date (Tues/Thurs)	Topics / Sections Covered
8/26 (Tues)	• Syllabus • 1.1 Angles • 1.2 Angle Relationships & Similar Triangles
8/28 (Thurs)	• 1.3 Trigonometric Functions • 1.4 Using Trigonometric Functions
9/2 (Tues)	• 2.1 Trig Functions of Acute Angles • 2.2 Trig Functions of Non-Acute Angles
9/4 (Thurs)	• 2.3 Using a Calculator to Find Trig Values • 2.4 Solving Right Triangles
9/9 (Tues)	• 2.5 Applications of Right Triangles, Bearings
9/11 (Thurs)	• 3.1 Radian Measure • 3.2 Arc Length & Area of a Sector
9/16 (Tues)	Exam 1 – Chapters 1 & 2
9/18 (Thurs)	• 3.3 Unit Circle & Circular Functions • 3.4 Linear & Angular Speeds
9/23 (Tues)	• 4.1 Graphs of Sine & Cosine Functions
9/25 (Thurs)	• 4.2 Translations of Sine & Cosine Graphs
9/30 (Tues)	• 4.3 Graphs of Tangent & Cotangent
10/2 (Thurs)	• 4.4 Graphs of Secant & Cosecant
10/7 (Tues)	• Section 4.4 Graphing HW Due (in class) • 5.1 Fundamental Identities
10/9 (Thurs)	Exam 2 – Chapters 3 & 4
10/14 (Tues)	• 5.2 Verifying Trig Identities • 5.3 Sum/Difference Identities for Cosine & Cofunctions (Part 1)
10/16 (Thurs)	• 5.3 Sum/Difference Identities for Cosine & Cofunctions (Part 2) • 5.4 Sum & Difference Identities for Sine & Tangent
10/21 (Tues)	• 5.5 Double-Angle Identities • 5.6 Half-Angle Identities
10/23 (Thurs)	• 6.1 Inverse Trig Functions • 6.2 Trig Equations I
10/28 (Tues)	• 6.3 Trig Equations II • 6.4 Equations with Inverse Trig Functions
10/30 (Thurs)	• Section 6.3 Additional HW Due (in class) • 7.1 Oblique Triangles & Law of Sines • Review for Exam 3
11/4 (Tues)	Exam 3 – Chapters 5 & 6
11/6 (Thurs)	• 7.2 Law of Sines – Ambiguous Case • 7.3 Law of Cosines
11/11 (Tues)	Veteran’s Day – Campus Closed
11/13 (Thurs)	• 7.4 Vectors
11/18 (Tues)	• 7.5 Applications of Vectors
11/20 (Thurs)	• 8.3 Complex Product & Quotient • 8.4 DeMoivre’s Theorem
11/25 (Tues)	• 8.5 Polar Equations & Graphs
11/27 (Thurs)	Thanksgiving – Campus Closed
12/2 (Tues)	• Q&A for Final Review
12/4 (Thurs)	• 8.1 Complex Numbers • 8.2 Trig Form of Complex Numbers
12/8 (Mon)	• 8.6 Parametric Equations, Graphs, Applications
12/10 (Wed)	Final Exam