Comprehensive Study Notes: Analytic and Applied Trigonometry for Precalculus

Analytic and Applied Trigonometry

Section 4.1: Right Triangle Trigonometry and Angle Measurement

This section covers the evaluation and application of trigonometric functions in right triangles, as well as the conversion and calculation of angle measures.

• Six Trigonometric Functions: Sine, cosine, tangent, cotangent, secant, and cosecant are defined for a right triangle as ratios of side lengths.

• Quadrant and Terminal Side: Determining the quadrant in which an angle's terminal side lies is essential for evaluating trigonometric functions.

• Degrees and Radians: Conversion between degrees and radians is fundamental. $\text{Radians} = \frac{\pi}{180} \times \text{Degrees}$

• Central Angles: Finding central angles from given angle measures in degrees or radians.

• Arc Length and Sector Area: Arc length: $s = r\theta$ (where $\theta$ is in radians) Sector area: $A = \frac{1}{2} r^2 \theta$

Section 4.2: The Unit Circle and Trigonometric Values

This section focuses on constructing and interpreting the unit circle, and evaluating trigonometric functions at key angles.

• Unit Circle Construction: The unit circle has a radius of 1 and is centered at the origin.

• Coordinates on the Unit Circle: For an angle $\theta$, the coordinates are $(\cos \theta, \sin \theta)$.

• Evaluating Trig Functions: Know the values for $0$,$\frac{\pi}{6}$,$\frac{\pi}{4}$,$\frac{\pi}{3}$,$\frac{\pi}{2}$, etc.

• Reference Angles: Use reference angles to evaluate trig functions for angles in different quadrants.

Section 4.3: Graphs and Transformations of Trigonometric Functions

This section addresses the graphical representation and transformation of trigonometric functions.

• Graphing Trig Functions: Plotting sine, cosine, and other trig functions, including amplitude, period, phase shift, and vertical shift.

• Domain and Range: Identifying the domain and range for each trig function.

• Transformations: Recognizing and applying transformations such as $y = a \sin(bx + c) + d$.

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

Section 4.4: Sine and Cosine Graphs

This section involves graphing sine and cosine functions, identifying key features, and determining their domains and ranges.

• Graphing Two Periods: Label maxima, minima, and midline.

• Equation or Graph: Relate the equation of a trig function to its graph.

• Domain and Range: For sine and cosine, domain is $(-\infty, \infty)$, range is $[-1, 1]$.

Section 4.5: Tangent and Cotangent Functions

This section covers the graphs and transformations of tangent and cotangent functions.

• Graphing: Draw approximate graphs of tangent and cotangent functions.

• Domain and Range: Tangent: domain excludes $\frac{\pi}{2} + k\pi$, range is $(-\infty, \infty)$.

• Transformations: Apply amplitude, period, and phase shift to tangent and cotangent graphs.

Section 4.6: Inverse Trigonometric Functions

This section introduces the inverse trigonometric functions and their properties.

• Inverse Values: Find $\sin^{-1} x$, $\cos^{-1} x$, $\tan^{-1} x$ for given values.

• Unit Circle Domain: Inverse trig functions are defined on restricted domains.

• Range of Inverse Functions: $\sin^{-1} x$ range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$; $\cos^{-1} x$ range: $[0, \pi]$; $\tan^{-1} x$ range: $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Section 5.1: Trigonometric Identities and Simplification

This section focuses on applying fundamental trigonometric identities to simplify expressions and solve equations.

• Reciprocal, Quotient, and Pythagorean Identities: $\sin \theta = \frac{1}{\csc \theta}$ $\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\sin^2 \theta + \cos^2 \theta = 1$

• Simplification: Use identities to rewrite and simplify trigonometric expressions.

• Algebraic Expansion: Solve trigonometric equations using algebraic techniques.

Section 5.2: Sum and Difference Formulas

This section covers the use of sum and difference formulas to evaluate trigonometric functions at non-standard angles.

• Sum and Difference Formulas: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

• Application: Use these formulas to find exact values for angles not on the unit circle.

Section 5.3: Double-Angle and Half-Angle Formulas

This section introduces formulas for evaluating trigonometric functions at double or half angles.

• Double-Angle Formulas: $\sin 2\theta = 2 \sin \theta \cos \theta$ $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$ $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

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

• Power-Reduction Formulas: Used to simplify expressions involving squared trig functions.

Section 5.4: Product-to-Sum and Sum-to-Product Formulas

This section covers formulas that convert products of trigonometric functions into sums or differences, and vice versa.

• Product-to-Sum Formulas: $\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$ $\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]$

• Sum-to-Product Formulas: $\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$

• Application: Use these formulas to simplify and solve trigonometric equations.

Section 5.5: Solving Trigonometric Equations

This section focuses on solving various types of trigonometric equations, including those involving multiple functions and quadratic forms.

• Linear-Type Equations: Solve equations like $a \sin x + b = 0$.

• Quadratic-Type Equations: Solve equations like $a \sin^2 x + b \sin x + c = 0$.

• Multiple Functions: Solve equations involving two different trig functions.

• General Solutions: Find all solutions in a given interval, such as $[0, 2\pi]$.

Unit Circle Values to Memorize

It is essential to memorize the following trigonometric values for common angles:

Angle	$\sin$	$\cos$	$\tan$
$0$	$0$	$1$	$0$
$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$\frac{\pi}{2}$	$1$	$0$	undefined

Additional info: Students should also be familiar with reciprocal and quotient identities, as well as the use of reference angles and symmetry in the unit circle.