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Chapter 5 Trigonometric Functions © 2011 Pearson Education, Inc. All rights reserved © 2010 2011 Pearson Education, Inc. All rights reserved 1 SECTION 5.5 Graphs of the Other Trigonometric Functions OBJECTIVES 1 2 Graph the tangent and cotangent functions. Graph the cosecant and secant functions. TANGENT FUNCTION The tangent function differs from the sine and cosine functions in three significant ways: 1. The tangent function has period π. 2. The tangent function is 0 when sin x = 0 and is undefined when cos x = 0. It is undefined at 3 5 , , , ... 2 2 2 3. The tangent function has no amplitude; that is, there are no minimum and maximum y-values. The range is (–∞, ∞). © 2011 Pearson Education, Inc. All rights reserved 3 GRAPH OF THE TANGENT FUNCTION Plotting y = tan x for common values of x and connecting the points with a smooth curve yields the following: © 2011 Pearson Education, Inc. All rights reserved 4 COTANGENT FUNCTION The cotangent function is similar to the tangent function: 1. The cotangent function has period π. 2. The cotangent function is 0 when cos x = 0 and is undefined when sin x = 0. It is undefined at , 3 , 5 , ... 3. The cotangent function has no amplitude; that is, there are no minimum and maximum y-values. The range is (–∞, ∞). © 2011 Pearson Education, Inc. All rights reserved 5 TANGENT AND COTANGENT FUNCTIONS Both functions are odd: tan (–x) = – tan x cot (–x) = – cot x 1 1 cot x and tan x tan 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). © 2011 Pearson Education, Inc. All rights reserved 6 GRAPH OF THE COTANGENT FUNCTION Using the information on the previous two slides, we graph y = cot x. © 2011 Pearson Education, Inc. All rights reserved 7 MAIN FACTS ABOUT y = tan x AND y = cot x y = tan x Period Domain y = cot x π π All real numbers All real numbers except odd except integer multiples of aa multiples of π 2 Range Vertical Asymptotes (–∞,∞) (–∞,∞) x = odd multiples of aa x = integer multiples of π 2 © 2011 Pearson Education, Inc. All rights reserved 8 MAIN FACTS ABOUT y = tan x AND y = cot x x-intercepts Symmetry y = tan x y = cot x At integer multiples of π At odd multiples of aa 2 tan (–x) = –tan x, cot (–x) = –cot x, odd function; odd function; so symmetric so symmetric with respect to with respect to the origin the origin © 2011 Pearson Education, Inc. All rights reserved 9 EXAMPLE 2 Graphing y = a cot [b(x – c)] Graph y 4 cot x over the interval 2 [–π, 2π]. Solution Step 1 For y 4 cot x , we have a = –4, 2 b = 1, and c 2 . Thus, vertical stretch factor = |–4| = 4 period 1 ; phase shift © 2011 Pearson Education, Inc. All rights reserved 2 10 EXAMPLE 2 Graphing y = a cot [b(x – c)] Solution continued Step 2 Locate two adjacent asymptotes. Solve x 2 0 x and 2 x . 2 3 x 2 3 Step 3 The interval , has length π. 2 2 3 The division points of , are: 2 2 © 2011 Pearson Education, Inc. All rights reserved 11 EXAMPLE 2 Graphing y = a cot [b(x – c)] Solution continued Step 3 continued 1 3 1 , , 2 4 4 2 2 3 5 . and 2 4 4 Step 4 Evaluate the function at those points: 3 x , 4 3 y 4cot 4 4 2 © 2011 Pearson Education, Inc. All rights reserved 12 EXAMPLE 2 Graphing y = a cot [b(x – c)] Solution continued Step 4 continued x , y 4cot 0 2 5 5 x , y 4cot 4 4 4 2 3 Step 5 Sketch vertical asymptotes: x , 2 2 Draw one cycle through the points above. Repeat the graph to the left and right over intervals of π. © 2011 Pearson Education, Inc. All rights reserved 13 EXAMPLE 2 Graphing y = a cot [b(x – c)] Solution continued y 4cot x 2 © 2011 Pearson Education, Inc. All rights reserved 14 COSECANT FUNCTION 1 Cosecant is the reciprocal of sine: csc x sin x 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. © 2011 Pearson Education, Inc. All rights reserved 15 GRAPH OF THE COSECANT FUNCTION The graphs of y = sin x and y = csc x over the interval [–2π, 2π] are shown to the right. © 2011 Pearson Education, Inc. All rights reserved 16 SECANT FUNCTION 1 Secant is the reciprocal of cosine: sec x cos x 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 3 5 , , , . At each of these points there is a 2 2 2 vertical asymptote. sec x = 1 when cos x = 1 and sec x = –1 when cos x = –1. © 2011 Pearson Education, Inc. All rights reserved 17 GRAPH OF THE SECANT FUNCTION The graphs of y = cos x and y = sec x over the interval 3 5 2 , 2 are shown to the right. © 2011 Pearson Education, Inc. All rights reserved 18 MAIN FACTS ABOUT y = csc x AND y = sec x y = csc x Period Domain y = sec x 2π 2π All real numbers All real numbers except integer except odd multiples of π multiples of aa 2 Range Vertical Asymptotes (–∞, –1] U [1, ∞) (–∞, –1] U [1, ∞) x = integer multiples of π © 2011 Pearson Education, Inc. All rights reserved x = odd multiples of aa 2 19 MAIN FACTS ABOUT y = csc x AND y = sec x x-intercepts Symmetry y = csc x y = sec x No x-intercepts No x-intercepts csc (–x) = –csc x, sec (–x) = sec x, odd function, even function origin symmetry with y-axis symmetry © 2011 Pearson Education, Inc. All rights reserved 20 EXAMPLE 4 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. © 2011 Pearson Education, Inc. All rights reserved 21 EXAMPLE 4 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. © 2011 Pearson Education, Inc. All rights reserved 22 EXAMPLE 4 Graphing a Range of Mach Numbers The Mach number, M, is given by speed of aircraft v x M M x csc , 2 speed of sound s 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 , ? 4 © 2011 Pearson Education, Inc. All rights reserved 23 EXAMPLE 4 Graphing a Range of Mach Numbers Solution x 1 x , first graph y sin . Because csc 2 sin x 2 2 Then use the reciprocal connection. © 2011 Pearson Education, Inc. All rights reserved 24 EXAMPLE 4 Graphing a Range of Mach Numbers Solution continued x For x , y csc csc 2.6. 4 2 8 x For x , y csc csc 1. 2 2 The range of Mach numbers associated with the interval , is (1,2.6]. 4 © 2011 Pearson Education, Inc. All rights reserved 25