Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphs of the Other Trigonometric Functions Learn to graph the tangent and cotangent functions. Learn to graph the cosecant and secant functions. SECTION

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TANGENT FUNCTION The tangent function differs form the sine and cosine function in three significant ways: 1.The tangent function has period π. 2.The tangent is 0 when sin x = 0 and is undefined when cos x = 0. It is undefined at 3.The tangent has no amplitude; no minimum and maximum y-values. Range is (–∞, ∞).

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GRAPH OF THE TANGENT FUNCTION If one were to make a table of values and plot

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley COTANGENT FUNCTION The cotangent function is similar to the tangent function: 1.The cotangent function has period π. 2.The cotangent is 0 when cos x = 0 and is undefined when sin x = 0. It is undefined at 3.The cotangent has no amplitude; no minimum and maximum y-values. Range is (–∞, ∞).

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TANGENT AND COTANGENT FUNCTIONS Both functions are odd: tan (–x) = – tan x cot (–x) = – cot x Both functions have the same sign everywhere they are both defined. When |tan x| is large, |cot x| is small, and conversely.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GRAPH OF THE COTANGENT FUNCTION Using the information on the previous two slides we graph y = cot x

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley MAIN FACTS ABOUT y = tan x and y = cot x y = tan xy = cot x Period π π Range (–∞, ∞) DomainAll real numbers except odd multiples of π/2. All real numbers except integer multiples of π. Vertical Asymptote x = a, where a is an odd multiple of π/2. x = a, where a is an integer multiple of π.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley MAIN FACTS ABOUT y = tan x and y = cot x y = tan xy = cot x x-interceptsa ± π/2, where a is an odd multiple of π/2. a ± π/2, where a is an integer multiple of π. Symmetrytan (–x) = –tan x odd function, symmetric with respect to the origin cot (–x) = –tan x odd function, symmetric with respect to the origin

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR GRAPHING y = a tan b(x – c) and y = a cot b(x – c) Step 1Find the vertical stretch factor = |a| and phase shift = c period = Step 2Locate two adjacent vertical asymptotes. For y = a tan b(x – c), solve For y = a cot b(x – c), solve

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR GRAPHING y = a tan b(x – c) and y = a cot b(x – c) Step 3Divide the interval on the x-axis between the two vertical asymptotes into 4 equal equal parts, each of length Step 4Evaluate the function at the three endpoints of the intervals found in Step 3 that re in the domain of the function.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR GRAPHING y = a tan b(x – c) and y = a cot b(x – c) Step 5Sketch the vertical asymptotes, using the values found in Step 2. Connect the points in Step 4 with a smooth curve in the standard shape of a cycle for the given function. Repeat the graph to the left and as needed.right over intervals of length

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Graphing y = a cot b(x – c) Graph over the interval [–π, 2π]. Thus, vertical stretch factor = |–4|; period phase shift Solution b = 1, and Forwe have a = –4, Step 1

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Graphing y = a cot b(x – c) Solution continued Step 2 Locate two adjacent asymptotes. Solve and Step 3 The intervalhas length π.The division points of are

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Graphing y = a cot b(x – c) Solution continued Step 3 continued Step 4 Evaluate the function at those points:

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Graphing y = a cot b(x – c) Solution continued Step 4 continued Draw one cycle through the points above. Repeat to the graph to the left and right over intervals of π. Step 5 Sketch vertical asymptotes:

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Graphing y = a cot b(x – c) Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley COSECANT FUNCTION Cosecant is the reciprocal of sine: Both functions have the same sign everywhere they are both defined. When |sin x| is large, |csc x| is small, and conversely. Csc x is undefined when sin x = 0. It is undefined at 0, ±π, ±2π, ±3π, …At each of these points there is a vertical asymptote. Csc x = 1 when sin x = 1 and csc x = –1 when sin x = –1.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GRAPH OF THE COSECANT FUNCTION The graphs of y = sin x and y = csc x over the interval [– 2π, 2π].

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SECANT FUNCTION Secant is the reciprocal of cosine: Both functions have the same sign everywhere they are both defined. When |cos x| is large, |sec x| is small, and conversely. Sec x is undefined when cos x = 0. It is undefined at ±π/2, ±3π/2, ±5π/2, … At each of these points there is a vertical asymptote. Sec x = 1 when cos x = 1 and sec x = –1 when cos x = –1.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GRAPH OF THE COSECANT FUNCTION The graphs of y = cos x and y = sec x over the interval

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley MAIN FACTS ABOUT y = csc x and y = sec x y = csc xy = sec x Period 2π DomainAll real numbers except integer multiples of π. All real numbers except odd multiples of π/2. Vertical Asymptote x = a, where a is an integer multiple of π. x = a, where a is an odd multiple of π/2. Range (–∞, –1] U [1, ∞)

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley MAIN FACTS ABOUT y = tan x and y = cot x y = csc xy = sec x x-interceptsNo x-intercepts. Symmetrycsc (–x) = –cscx odd function, symmetric with respect to the origin sec (–x) = sec x even function, symmetric with respect to the y-axis

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Graphing y = a csc b(x – c) Graph y = 3csc 2x over a two-period interval. Follow the steps as given in section 5.4. no phase shift since c = 0 amplitude = 3, Step 1 y = 3sin 2x Solution Because when sin 2x = ±1. First graph y = 3sin 2x.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Graphing y = a csc b(x – c) Solution continued Step 2 Starting point: x = 0. One cycle is [0, π]. Step 3

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Graphing y = a csc b(x – c) Solution continued Step 4 Sketch the graph of y = 3sin 2x through the points (0, 0) Use the reciprocal relationship to graph y = 3csc 2x, starting at the common points. Step 5 Extend the graph to interval

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Graphing y = a csc b(x – c) Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Graphing a Range of Mach Numbers When a plane travels at supersonic and hypersonic speeds, small disturbances in the atmosphere are transmitted downstream within a cone. The cone intersects the ground, and the edge of the cone’s intersection with the ground can be represented as in the figure on the next slide. The sound waves strike the edge of the cone at a right angle. The speed of the sound wave is represented by leg s of the right triangle shown in the figure.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Graphing a Range of Mach Numbers The plane is moving at speed v, which is represented by the hypotenuse of the right triangle in figure.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Graphing a Range of Mach Numbers The Mach number, M, is given by where x is the angle of the vertex of the cone. Graph the Mach number function, M(x), as the angle at the vertex of the cone varies. What is the range of Mach numbers associated with the interval

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Graphing a Range of Mach Numbers Solution Because first graph Then use the reciprocal connection.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Graphing a Range of Mach Numbers Solution continued interval The range of Mach numbers associated with the is (1, 2.6].