Precalculus Study Notes: Trigonometric Functions and Their Graphs

Orientation & Introduction to Trigonometric Functions

Course Overview and Resources

This section introduces the structure of the Precalculus course, focusing on trigonometric functions. Students are encouraged to utilize online resources such as MyLabMath and eText for supplementary learning.

• Course Design: Understanding the flow and expectations of the course.

• Resources: Access to digital textbooks and practice platforms.

Angles: Degree and Radian Measure

Angles are fundamental in trigonometry and can be measured in degrees or radians.

• Degree: A unit of angular measure; a full circle is 360 degrees.

• Radian: The standard unit of angular measure in mathematics; a full circle is $2\pi$ radians.

• Conversion Formula: $\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

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

Applications of Radian Measure

Arc Length and Sector Area

Radian measure is used to calculate arc lengths and areas of sectors in circles.

• Arc Length Formula: $s = r\theta$ (where $s$ is arc length, $r$ is radius, $\theta$ is angle in radians)

• Sector Area Formula: $A = \frac{1}{2} r^2 \theta$

• Example: For a circle of radius 5 and angle $\frac{\pi}{3}$ radians, arc length $s = 5 \times \frac{\pi}{3} = \frac{5\pi}{3}$

Triangles and Right Triangle Trigonometry

Right Triangle Trigonometry

Trigonometric ratios relate the angles and sides of right triangles.

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

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

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

• Example: In a triangle with sides 3, 4, 5, $\sin \theta = \frac{3}{5}$

Trigonometric Functions of General Angles

Trigonometric functions can be extended to any angle, not just those in right triangles.

• Reference Angle: The acute angle formed by the terminal side of the angle and the x-axis.

• Signs in Quadrants: Sine, cosine, and tangent have different signs depending on the quadrant.

• Example: $\sin(150^\circ) = \sin(30^\circ) = 0.5$ (since 150° is in the second quadrant where sine is positive)

The Unit Circle

Definition and Properties

The unit circle is a circle of radius 1 centered at the origin, used to define trigonometric functions for all angles.

• Coordinates: Any point on the unit circle has coordinates $(\cos \theta, \sin \theta)$

• Key Angles: $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$

• Example: At $\theta = \frac{\pi}{2}$, the point is $(0, 1)$

Graphs of Trigonometric Functions

Graphs of Sine and Cosine

Sine and cosine functions produce periodic wave-like graphs.

• 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 $2\pi$

Graphs of Tangent, Cotangent, Secant, and Cosecant

These functions have distinct graphs with vertical asymptotes and periodicity.

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

• Cotangent: $y = \cot x$ has period $\pi$ and vertical asymptotes at $x = k\pi$

• Secant: $y = \sec x$ is the reciprocal of cosine, undefined where $\cos x = 0$

• Cosecant: $y = \csc x$ is the reciprocal of sine, undefined where $\sin x = 0$

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles from known trigonometric values.

• Arcsin: $y = \arcsin x$

• Arccos: $y = \arccos x$

• Arctan: $y = \arctan x$

• Domain and Range: Each inverse function has specific domain and range restrictions.

• Example: $\arcsin(0.5) = \frac{\pi}{6}$

Trigonometric Identities

Fundamental Identities

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

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

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

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

• Example: If $\sin x = 0.6$, then $\cos x = \sqrt{1 - 0.36} = 0.8$

Summary Table: Trigonometric Functions and Their Properties

Function	Period	Domain	Range	Key Features
Sine ($\sin x$)	$2\pi$	All real numbers	[$-1$, $1$]	Wave, amplitude 1
Cosine ($\cos x$)	$2\pi$	All real numbers	[$-1$, $1$]	Wave, amplitude 1
Tangent ($\tan x$)	$\pi$	All real numbers except $x \neq \frac{\pi}{2} + k\pi$	All real numbers	Vertical asymptotes
Cotangent ($\cot x$)	$\pi$	All real numbers except $x \neq k\pi$	All real numbers	Vertical asymptotes
Secant ($\sec x$)	$2\pi$	All real numbers except $x \neq \frac{\pi}{2} + k\pi$	($-\infty$$,$] $\cup$ [$1$,$\infty$)	Vertical asymptotes
Cosecant ($\csc x$)	$2\pi$	All real numbers except $x \neq k\pi$	($-\infty$$,$] $\cup$ [$1$,$\infty$)	Vertical asymptotes

Midterm Review

Preparation and Practice

Students should review all topics, practice problems, and understand key formulas and identities for the midterm exam.

• Review: Go over quizzes, tests, and homework assignments.

• Practice: Solve problems involving angle conversion, trigonometric ratios, and graphing functions.

• Understand: Be able to apply identities and solve equations involving trigonometric functions.