MAT 110: Algebra and Trigonometry - Core Study Guide

Course Overview

This study guide summarizes the main topics and learning outcomes for MAT 110: Algebra and Trigonometry, based on the syllabus and assignments from College Algebra & Trigonometry, 6th Ed. by Dugopolski. The course emphasizes both algebraic and trigonometric concepts, focusing on functions, graphs, trigonometric functions, identities, and applications such as the Law of Sines and Law of Cosines.

Functions and Graphs

Functions

Functions are fundamental mathematical objects that describe relationships between sets. Understanding functions is essential for modeling and solving real-world problems.

• Definition: A function is a relation that assigns exactly one output value to each input value from a domain.

• Notation: $f(x)$ denotes the function value at $x$.

• Domain and Range: The domain is the set of possible input values; the range is the set of possible output values.

• Example: $f(x) = x^2$ has domain $(-\infty, \infty)$ and range $[0, \infty)$.

Graphs of Relations and Functions

Graphing functions helps visualize their behavior and properties.

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

• Example: The graph of $y = x^2$ passes the vertical line test.

Families of Functions, Transformations, and Symmetry

Functions can be grouped into families and transformed to create new functions.

• Transformations: Include translations, reflections, stretches, and compressions.

• Symmetry: Functions may be even ($f(-x) = f(x)$), odd ($f(-x) = -f(x)$), or neither.

• Example: $f(x) = |x|$ is even; $f(x) = x^3$ is odd.

Operations with Functions

Functions can be combined using arithmetic operations.

• Addition: $(f+g)(x) = f(x) + g(x)$

• Subtraction: $(f-g)(x) = f(x) - g(x)$

• Multiplication: $(fg)(x) = f(x) \cdot g(x)$

• Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Composition: $(f \circ g)(x) = f(g(x))$

• Example: If $f(x) = x+1$ and $g(x) = x^2$, then $(f \circ g)(x) = f(g(x)) = x^2 + 1$.

Inverse Functions

An inverse function reverses the effect of the original function.

• Definition: $f^{-1}(x)$ is the inverse of $f(x)$ if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x-3}{2}$.

The Trigonometric Functions

Angles and Their Measurements

Angles are measured in degrees and radians, which are essential for trigonometric calculations.

• Degree: One full rotation is $360^\circ$.

• Radian: One full rotation is $2\pi$ radians.

• Conversion: $1^\circ = \frac{\pi}{180}$ radians

• Example: $90^\circ = \frac{\pi}{2}$ radians

The Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions describing ratios in right triangles and periodic phenomena.

• Definitions: For angle $\theta$ in a right triangle:

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

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

• Unit Circle: $\sin \theta$ and $\cos \theta$ correspond to $y$ and $x$ coordinates on the unit circle.

• Example: $\sin 30^\circ = \frac{1}{2}$, $\cos 60^\circ = \frac{1}{2}$

The Graphs of the Sine and Cosine Functions

The graphs of sine and cosine are periodic and have amplitude, period, and phase shift.

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

• Period: $\frac{2\pi}{B}$

• Amplitude: $|A|$

• Example: $y = 2 \sin(x)$ has amplitude $2$and period $2\pi$.

The Other Trigonometric Functions and Their Graphs

Other trigonometric functions include tangent, cotangent, secant, and cosecant.

• Definitions:

  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$

  • $\cot \theta = \frac{\cos \theta}{\sin \theta}$

  • $\sec \theta = \frac{1}{\cos \theta}$

  • $\csc \theta = \frac{1}{\sin \theta}$

• Graphs: These functions have vertical asymptotes where their denominators are zero.

• Example: $\tan x$ has period $\pi$ and vertical asymptotes at $x = \frac{\pi}{2} + k\pi$.

The Inverse Trigonometric Functions

Inverse trigonometric functions allow finding angles from given ratios.

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

• Example: $\sin^{-1}(\frac{1}{2}) = 30^\circ$ or $\frac{\pi}{6}$ radians

Right Triangle Trigonometry

Trigonometric functions are used to solve right triangles.

• Key Formulas:

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

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

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

• Example: In a triangle with sides $3$,$, $5$, $\sin \theta = \frac{3}{5}$if $3$ is opposite $\theta$.

Trigonometric Identities and Conditional Equations

Basic Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values in their domains.

• Pythagorean Identities:

  • $\sin^2 \theta + \cos^2 \theta = 1$

  • $1 + \tan^2 \theta = \sec^2 \theta$

  • $1 + \cot^2 \theta = \csc^2 \theta$

• Reciprocal Identities:

  • $\sin \theta = \frac{1}{\csc \theta}$

  • $\cos \theta = \frac{1}{\sec \theta}$

  • $\tan \theta = \frac{1}{\cot \theta}$

Verifying Identities

Verifying identities involves transforming one side of an equation to match the other using algebraic and trigonometric properties.

• Strategy: Use known identities, simplify expressions, and factor where possible.

• Example: Verify $\frac{1-\cos^2 \theta}{\sin^2 \theta} = 1$ using $\sin^2 \theta + \cos^2 \theta = 1$.

Sum and Difference Identities

These identities express trigonometric functions of sums or differences of angles.

• Formulas:

  • $\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$

  • $\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$

  • $\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$

• Example: $\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$

Double-Angle and Half-Angle Identities

These identities relate trigonometric functions of double or half angles to those of the original angle.

• Double-Angle:

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

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

  • $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

• Half-Angle:

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

  • $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$

Product-Sum and Sum-Product Identities

These identities convert products of sines and cosines into sums or vice versa.

• Product-Sum:

  • $\sin a \sin b = \frac{1}{2}[\cos(a-b) - \cos(a+b)]$

  • $\cos a \cos b = \frac{1}{2}[\cos(a-b) + \cos(a+b)]$

  • $\sin a \cos b = \frac{1}{2}[\sin(a+b) + \sin(a-b)]$

Conditional Trigonometric Equations

Conditional equations are trigonometric equations that are true for specific values of the variable.

• Solving: Use identities and algebraic methods to find all solutions within a given interval.

• Example: Solve $\sin x = \frac{1}{2}$ for $x$ in $[0, 2\pi]$; solutions are $x = \frac{\pi}{6}, \frac{5\pi}{6}$.

Law of Sines and Law of Cosines

Law of Sines

The Law of Sines relates the sides and angles of any triangle, not just right triangles.

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

• Application: Used to solve triangles when two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known.

• Example: If $A = 30^\circ$, $B = 45^\circ$, $a = 10$, find $b$ using $\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}$.

Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem for any triangle.

• Formula: $c^2 = a^2 + b^2 - 2ab \cos C$

• Application: Used when two sides and the included angle (SAS) or all three sides (SSS) are known.

• Example: If $a = 5$, $b = 7$, $C = 60^\circ$, then $c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cos 60^\circ$.

Course Grading Table

The following table summarizes the grading scale for MAT 110:

Average (%)	Grade
93 - 100	A
90 - 92.9	A-
88 - 89.9	B+
83 - 87.9	B
80 - 82.9	B-
78 - 79.9	C+
70 - 77.9	C
60 - 69.9	D
Below 60	F

Summary of Assignments

Assignments are based on textbook sections and include exercises for practice and mastery. Students are expected to complete homework for each section, though it is not collected. Quizzes and tests are scheduled with firm deadlines.

Academic Integrity and Support

Students are expected to uphold academic integrity and complete their own work. Free tutoring and academic support are available at the Academic Support Center.

Technical Requirements

Success in the online course requires proficiency with the learning management system, email, file creation and submission, and use of a calculator (TI-83 recommended).