Trig functions and x and y. Elementary Functions. The tangent function. The tangent function

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1 Trig functions and x and y In this presentation we describe the graphs of each of the six trig functions. We have already focused on the sine and cosine functions, devoting an entire lecture to the sine wave. Now we look at the tangent function and then the reciprocals of sine, cosine and tangent, that is, cosecant, secant and cotangent. Part 4, Trigonometry Lecture 4.5a, Graphing Trig Functions First a note about notation. Up to this time we have viewed trig functions as functions of an angle θ and have tended to reserve the letters x, y for coordinates on the unit circle. But it is time to return to our original custom about variables in functions, using x as the input variable and y as the output variable. For example, when we write y = tan(x) we now think of x as an angle and y as a ratio of two sides of a triangle. (In this case x is the old θ and y is the old xy!) Dr. Ken W. Smith Sam Houston State University 1 / The tangent function / 4 / The tangent function sin x The tangent function tan x = cos x has a zero wherever sin x = 0, that is, whenever x is..., π, π, 0, π,..., πk,... (where k is an integer.) Here is the graph of the tangent function. The tangent function is undefined whenever cos x = 0, that is, at the (k+1)π π π 3π 5π x-values..., 3π,... (where k is an integer.),,,,,..., Indeed, at these x-values, the tangent function has vertical asymptotes. 3 /

2 The tangent function Central Angles and Arcs When we discussed the sine wave, we also discussed concepts of period, amplitude and phase shift. The graph of y = sin x has period π, amplitude 1 and phase shift 0. We observed earlier that the tangent function has period π. This is clear from the unit circle definition of tangent and this period is visible in our graph. It does not make sense to discuss the amplitude of the tangent function since the range of tangent is the full set of all real numbers, (, ). The domain of the tangent function is all real numbers except those where cos x = 0. We can write this in set notation as... ( 3π, π ) ( π, π ) (π, 3π ) (3π, 5π ).... Since this domain is a union of an infinite number of open intervals (each interval of length π) then we might write this union in a more compact form using a more general iterated union notation: Domain of the tangent function = k= ( (k 1) π, (k + 1) π). (We won t do much with these more general arbitrary unions in this class, but it is important to see this notation once or twice in a precalculus class.) 5 / The graphs of secant and cosecant The secant function is the reciprocal of cosine and so it has vertical asymptotes wherever cos x = 0. Here is the graph of the secant function (in blue) with asymptotes as dotted red lines and the cosine function hiding in light yellow. 6 / The graphs of secant and cosecant Since 1 cos x 1 then the reciprocal function, secant, is bounded away from the x-axis; whenever cos x is positive (but no larger than 1) then the secant is positive but greater than or equal to 1. Similarly whenever the cosine is negative (but not less than 1) the secant function is negative but less than or equal to 1. 7 / 8 /

3 The graphs of secant and cosecant The graphs of secant and cosecant The graph of the cosecant function is similar to the graph of the secant function. The cosecant function is the reciprocal of the sine function. When we investigated the sine and cosine functions we observed that the cosine function is the sine function shifted to the left by π (that is, cos x = sin(x + π )) and so the graph of the sine function is the same as the graph of the cosine function shifted to the right by π. If the graph of sine is achieved by shifting cosine to the right by π then the graph of cosecant is the secant function shifted to the right by π. 9 / The cotangent function The cotangent is the reciprocal of tangent. Here is the graph of the cotangent function. 10 / The cotangent function The cotangent is the reciprocal of tangent. We see from looking at the graph of cotangent that the graph of cotangent can be achieved by taking the graph of the tangent function, moving it left (or right) by π and then reflecting it across the x-axis. cot(x) = tan(x + π ). 11 / 1 /

4 Tangent and cotangent Tangent and cotangent Another way to look at the cotangent function: since cos x = sin(x + π ) and that sin x = sin(x + π) then sin(x + π ) sin(x + π ) sin(x + π ) cos x π cot(x) = = = = π = tan(x+ ). sin x sin x sin(x + π) cos(x + ) 13 / 14 / Graphs of the six trig functions In the next presentation, we work through some exercises with the graphs of the six trig functions. Part 4, Trigonometry Lecture 4.5b, Graphing Trig Functions: Some Worked Problems (End) Dr. Ken W. Smith Sam Houston State University 15 / 16 /

5 Some worked problems Some worked problems For each of the following functions, describe the transformation required to change the graph of the tangent function into the graph of the indicated function. 1 y = tan(x π ) y = tan(x π ) Solutions. 1 To graph y = tan(x π ), shift the graph of the tangent function right by π. To graph y = tan(x π ) = tan((x π 4 )), shift the graph of the tangent function right by π 4 and then shrink the function by a factor of two in the horizontal direction (centered about the line x = π 4.) For each of the following functions, describe the transformation required to change the graph of the tangent function into the graph of the indicated function. 3 y = 5 tan(x π ) y = tan(x π ) + 4 Solutions. 3 To graph y = 5 tan(x π ) + 1, shift the graph of the tangent function right by π, stretch it vertically by a factor of 5 and then move the function up 1. 4 To graph y = tan(x π ) + 4 = tan((x π 4 )) + 4, shift the graph of the tangent function right by π 4, then shrink the function by a factor of two in the horizontal direction, stretch it by a factor of in the vertical direction, reflect it across the x-axis, and then shift it up by / Some worked problems For each of the following functions, describe the transformation required to change the graph of the tangent function into the graph of the indicated function. 5 y = cot(x) 6 y = cot(x π ) 7 y = cot(x π ) Solutions. 5 Since cot(x) = tan(x + π ) then to graph y = cot(x), reflect the graph of y = tan x across the x-axis and shift it left by π. 6 To graph y = cot(x π ), first reflect the graph of y = tan x across the x-axis and shift it left by π to obtain the graph of the cotangent function. Finally, shift the graph right by π. 7 To graph y = cot(x π ), first reflect the graph of y = tan x across the x-axis and shift it left by π to obtain the graph of the cotangent function. Then shift the graph right by π 4 and then shrink the function by a factor of two in the horizontal direction. 19 / 18 / Two more worked problems 8 Find all solutions to the trig equation tan θ = 1 Solution. From looking at the unit circle, we see that θ = 45 = π/4 is a solution to this equation. So also is θ = 5 = 5π/4, the angle in the third quadrant with reference angle π/4. But there are many more solutions; if we add π to θ, we get new angles that satisfy this equation. Therefore { π 4 + πk : k Z} {5π 4 is the (infinite) set of all solutions. + πk : k Z} However, recall that the tangent function has period π. So we could simplify this answer by just writing { π 4 + πk : k Z} 0 /

6 Two more worked problems Worked problems on graphing trig functions 9 The angle θ has the property that sec θ = and tan θ is negative. Identify the angle θ and then find all six trig functions of the angle θ. Solution. Since the secant of θ is then cos(θ) = 1. Since the tangent of θ is negative then θ is in the fourth quadrant and we may assume θ = 30 = π 3 3. Then sin(θ) = and tan(θ) = 3 and the other functions are reciprocals of these. In the next presentation, we will look at inverse trig functions, that is, the inverse functions of cosine, sin, tangent, secant, cosecant and cotangent. (End) 1 / /