Trigonometric Graphs Overview

Overview

This lecture covers how to graph the secant, cosecant, tangent, and cotangent trigonometric functions, focusing on their properties, transformations, and the relationships with their base sine and cosine graphs.

Graphing Cosecant and Secant Functions

• Cosecant (csc x) is the reciprocal of sine: csc x = 1/sin x.
• Secant (sec x) is the reciprocal of cosine: sec x = 1/cos x.
• Where sine or cosine is zero, cosecant and secant have vertical asymptotes (undefined).
• The max/min of sine and cosine correspond to the min/max of cosecant and secant.
• Both secant and cosecant have a period of 2π and no amplitude since their graphs are unbounded.
• Cosecant is an odd function (symmetric about the origin); secant is even (symmetric about the y-axis).

Example: Transformations of Cosecant

• For y = –2 csc(3x), the graph reflects across the x-axis and has amplitude 2.
• The period is 2π/3; one cycle occurs between 0 and 2π/3.
• Vertical asymptotes occur at even multiples of π/3.
• For y = –2 csc(3x) + 1, the entire graph is shifted up by 1 unit.

Graphing Tangent and Cotangent Functions

• Tangent (tan x) = sin x / cos x; x-intercepts where sin x = 0, vertical asymptotes where cos x = 0.
• Tangent function is odd with a period of π (one cycle between two vertical asymptotes).
• Cotangent (cot x) = cos x / sin x; x-intercepts where cos x = 0, vertical asymptotes where sin x = 0.
• Cotangent function is also odd, symmetric about the origin, period π.

Example: Transformations of Tangent

• For y = tan(4x), the period is π/4 (period = π/B).
• X-intercepts occur at the midpoints between vertical asymptotes.
• For y = 4 tan(2x + π/3), solve 2x + π/3 between –π/2 and π/2 for one period.
• Vertical asymptotes and x-intercepts are found using interval midpoints and solving inequalities.
• For y = –tan(3x)/4 + 2, the graph reflects over the x-axis and shifts up by 2 units.

Key Terms & Definitions

• Cosecant (csc x) — 1 divided by sine x (1/sin x).
• Secant (sec x) — 1 divided by cosine x (1/cos x).
• Vertical Asymptote — Line where the function is undefined due to division by zero.
• Period — The interval after which the function repeats its values.
• Odd Function — Symmetric about the origin.
• Even Function — Symmetric about the y-axis.
• Amplitude — Half the distance between max and min (not applicable to secant/cosecant).

Action Items / Next Steps

• Practice graphing transformations of secant, cosecant, tangent, and cotangent.
• Solve for periods, asymptotes, and x-intercepts for assigned trigonometric function equations.