**Graphs of Other Trigonometric Functions** MATH 109 - Precalculus S. Rook

**Overview** • Section 4.6 in the textbook: • Graphing the tangent & cotangent functions • Graphing the secant & cosecant functions

**Graphing the Tangent & Cotangent Functions**

**Graph of the Parent Tangent Function y = tan x** • Recall that on the unit circle any point (x, y) can be written as (cos θ, sin θ) • It can be shown that the period of y = tan x is π • By examining the unit circle • Thus, by taking y⁄x of each point on the circumference of the unit circle we generate one cycle of y = tan x, 0 < x < π

**Graph of the Parent Tangent Function y = tan x (Continued)** • To graph any tangent function we need to know: • A set of points on the parent function y = tan x • (0, 0), (π⁄4, 1), (π⁄2 , und), (3π⁄4, -1), (π, 0) • Naturally these are not the only points, but are often the easiest to manipulate • The shape of the graph

**Graph of the Parent Tangent Function y = cot x** • Recall that on the unit circle any point (x, y) can be written as (cos θ, sin θ) • It can be shown that the period of y = cot x is π • By examining the unit circle • Thus, by taking x⁄y of each point on the circumference of the unit circle we generate one cycle of y = cot x, 0 < x < π

**Graph of the Parent Cotangent Function y = cot x (Continued)** • To graph any tangent function we need to know: • A set of points on the parent function y = tan x • (0, und), (π⁄4, 1), (π⁄2 , 0), (3π⁄4, -1), (π, und) • Naturally these are not the only points, but are often the easiest to manipulate • The shape of the graph

**Graphing y = d + a tan(bx + c) or y = d + a cot(bx + c)** • To graph y = d + a tan(bx + c) or y = d + a cot(bx + c) : • Establish the y-axis • Establish the x-axis • The x-values of the 5 points in the are the transformed x-values for the final graph • Use transformations to calculate the y-values for the final graph • Connect the points in the shape of the tangent or cotangent – this is 1 cycle • Be aware of reflection when it exists • Extend the graph if necessary

**Graphing the Tangent & Cotangent Functions** Ex 1: Graph by identifying i) period, phase shift, and vertical translation ii) extend the graph one period forwards a) b)

**Graphing the Secant & Cosecant Functions**

**Graph of the Parent Secant Function y = sec x** • Recall that on the unit circle any point (x, y) can be written as (cos θ, sin θ) • Period of y = sec x is 2π • Reciprocal of y = cos x • Thus, by taking 1⁄x of each point on the circumference of the unit circle we generate one cycle of y = sec x, 0 < x < 2π

**Graph of the Parent Secant Function y = sec x (Continued)** • To graph any secant function we need to know: • A set of points on the parent function y = tan x • (0, 1), (π⁄2, und), (π, -1), (3π⁄2, und), (2π, 1) • Naturally these are not the only points, but are often the easiest to manipulate • The shape of the graph

**Graph of the Parent Cosecant Function y = csc x** • Recall that on the unit circle any point (x, y) can be written as (cos θ, sin θ) • Period of y = csc x is 2π • Reciprocal of y = sin x • Thus, by taking 1⁄y of each point on the circumference of the unit circle we generate one cycle of y = csc x, 0 < x < 2π

**Graph of the Parent Cosecant Function y = csc x (Continued)** • To graph any cosecant function we need to know: • A set of points on the parent function y = csc x • (0, und), (π⁄4, 1), (π⁄2 , 0), (3π⁄4, -1), (π, und) • Naturally these are not the only points, but are often the easiest to manipulate • The shape of the graph

**Graphing y = d + a sec(bx + c) or y = d + a csc(bx + c)** • To graph y = d + a sec(bx + c) or y = d + a csc(bx + c) : • Establish the y-axis • Establish the x-axis • The x-values of the 5 points in the are the transformed x-values for the final graph • Use transformations to calculate the y-values for the final graph • Connect the points in the shape of the secant or cosecant– this is 1 cycle • Be aware of reflection when it exists • Extend the graph if necessary

**Graphing the Secant & Cosecant Functions** Ex 2: Graph by identifying i) period, phase shift, and vertical translation ii) extend the graph one period backwards a) b)

**Summary** • After studying these slides, you should be able to: • Graph tangent & cotangent functions • Graph secant & cosecant functions • Additional Practice • See the list of suggested problems for 4.6 • Next lesson • Inverse Trigonometric Functions (Section 4.7)