Trigonometric Graphs and Properties

Overview

This lecture explains how to graph the main trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—focusing on their periods, amplitudes, phase/vertical shifts, and key properties.

Graphs and Properties of Sine and Cosine Functions

• The domain of y = sin(x) and y = cos(x) is all real numbers; their range is [–1, 1].
• Cosine is an even function (cos(–x) = cos(x)); sine is odd (sin(–x) = –sin(x)).
• Both functions are periodic with period 2π, continuous, and smooth.
• The fundamental cycle of cos(x) and sin(x) occurs on [0, 2π].
• The graph of sin(x) is a right-shift of cos(x) by π/2 units.
• Amplitude is the height from the midline to a peak (1 for basic graphs); vertical and phase shifts move the graph up/down and left/right.
• The general form y = A sin(Bx + C) + D or y = A cos(Bx + C) + D allows for amplitude (|A|), period (2π/|B|), phase shift (–C/B), and vertical shift (D).

Graphs of Secant and Cosecant Functions

• Secant and cosecant are reciprocals of cosine and sine, respectively.
• The domain of sec(x): all real x except x = (2k+1)π/2, k ∈ ℤ; range: (–∞, –1] ∪ [1, ∞).
• The domain of csc(x): all real x except x = kπ, k ∈ ℤ; range: (–∞, –1] ∪ [1, ∞).
• Both have period 2π, are continuous and smooth in their domains, and have vertical asymptotes where their denominator function is zero.
• No amplitude is defined, since secant and cosecant are unbounded.

Graphs of Tangent and Cotangent Functions

• Tangent and cotangent have domains excluding values where the denominator is zero.
• The period of tan(x) and cot(x) is π; range is (–∞, ∞).
• Tangent is undefined at x = (2k+1)π/2; cotangent is undefined at x = kπ.
• Both graphs have vertical asymptotes and are continuous/smooth in their domains.
• Both are odd functions.

Sinusoids and Transformations

• Any function of the form y = A sin(Bx + C) + D or y = A cos(Bx + C) + D is a sinusoid.
• Sinusoids are characterized by amplitude, period, phase shift, and vertical shift.
• Phase shift is determined by –C/B; period is 2π/|B|; amplitude is |A|; vertical shift is D.
• Identities can convert between sine and cosine forms and help rewrite mixed trigonometric expressions as sinusoids.

Key Terms & Definitions

• Period — Smallest positive value p such that f(x + p) = f(x) for all x.
• Amplitude — Half the distance between the maximum and minimum values of a sinusoid.
• Phase Shift — Horizontal shift of a graph; for y = sin(Bx + C), it’s –C/B.
• Vertical Shift — Up/down movement of the entire graph; value D in y = ... + D.
• Sinusoid — Any function in the form y = A sin(Bx + C) + D or y = A cos(Bx + C) + D.
• Even Function — f(–x) = f(x); e.g., cosine.
• Odd Function — f(–x) = –f(x); e.g., sine and tangent.

Action Items / Next Steps

• Complete exercises: Graph one cycle for each given function, identify period, amplitude, phase and vertical shift.
• Practice rewriting mixed trigonometric functions into sinusoidal forms as shown in the lecture.
• Review sum and cofunction identities for converting between trigonometric forms.