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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. Elementary Functions 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 2013 Smith (SHSU) Elementary Functions 2013 1 / 22 The tangent function Smith (SHSU) Elementary Functions 2013 2 / 22 2013 4 / 22 The tangent function sin x The tangent function tan x = cos x has a zero wherever sin x = 0, that is, whenever x is . . . , −2π, − 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 (2k+1)π π π 3π 5π x-values . . . , − 3π , ... (where k is an integer.) 2 , − 2, 2, 2 , 2 ,..., 2 Indeed, at these x-values, the tangent function has vertical asymptotes. Smith (SHSU) Elementary Functions 2013 3 / 22 Smith (SHSU) Elementary Functions The tangent function Central Angles and Arcs When we discussed the sine wave, we also discussed concepts of period, amplitude and phase shift. The domain of the tangent function is all real numbers except those where cos x = 0. The graph of y = sin x has period 2π, amplitude 1 and phase shift 0. We can write this in set notation as 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, (−∞, ∞). . . . (− π π π 3π 3π 5π 3π π , − ) ∪ (− , ) ∪ ( , ) ∪ ( , ) . . . . 2 2 2 2 2 2 2 2 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=−∞ ( (2k − 1) (2k + 1) π, π). 2 2 (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.) Smith (SHSU) Elementary Functions 2013 5 / 22 Smith (SHSU) Elementary Functions 2013 The graphs of secant and cosecant The graphs of secant and cosecant The secant function is the reciprocal of cosine and so it has vertical asymptotes wherever cos x = 0. 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. Here is the graph of the secant function (in blue) with asymptotes as dotted red lines and the cosine function hiding in light yellow. Smith (SHSU) Elementary Functions 2013 6 / 22 Similarly whenever the cosine is negative (but not less than −1) the secant function is negative but less than or equal to −1. 7 / 22 Smith (SHSU) Elementary Functions 2013 8 / 22 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 π2 (that is, cos x = sin(x + π2 )) and so the graph of the sine function is the same as the graph of the cosine function shifted to the right by π2 . If the graph of sine is achieved by shifting cosine to the right by π2 then the graph of cosecant is the secant function shifted to the right by π2 . Smith (SHSU) Elementary Functions 2013 9 / 22 Smith (SHSU) Elementary Functions 2013 10 / 22 The cotangent function The cotangent function The cotangent is the reciprocal of tangent. Here is the graph of 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 π2 and then reflecting it across the x-axis. π cot(x) = − tan(x + ). 2 Smith (SHSU) Elementary Functions 2013 11 / 22 Smith (SHSU) Elementary Functions 2013 12 / 22 Tangent and cotangent Tangent and cotangent Another way to look at the cotangent function: since cos x = sin(x + π2 ) and that − sin x = sin(x + π) then sin(x + π2 ) sin(x + π2 ) sin(x + π2 ) cos x π cot(x) = = =− =− π = − tan(x+ ). sin x sin x sin(x + π) cos(x + 2 ) 2 Smith (SHSU) Elementary Functions 2013 13 / 22 Smith (SHSU) Elementary Functions 2013 14 / 22 Graphs of the six trig functions Elementary 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 2013 Smith (SHSU) Elementary Functions 2013 15 / 22 Smith (SHSU) Elementary Functions 2013 16 / 22 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 − π2 ) 2 y = tan(2x − π2 ) 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 − π2 ) + 1 4 y = −2 tan(2x − π2 ) + 4 Solutions. Solutions. 1 2 3 To graph y = tan(x − π2 ), shift the graph of the tangent function right by π2 . To graph y = tan(2x − π2 ) = tan(2(x − π4 )), shift the graph of the tangent function right by π4 and then shrink the function by a factor of two in the horizontal direction (centered about the line x = π4 .) Smith (SHSU) Elementary Functions 2013 17 / 22 Some worked problems To graph y = −2 tan(2x − π2 ) + 4 = −2 tan(2(x − π4 )) + 4, shift the graph of the tangent function right by π4 , then shrink the function by a factor of two in the horizontal direction, stretch it by a factor of 2 in the vertical direction, reflect it across the x-axis, and then shift it up by 4. Smith (SHSU) Elementary Functions 2013 18 / 22 Two more 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) π 6 y = cot(x − 2 ) π 7 y = cot(2x − 2 ) Solutions. π 5 Since cot(x) = − tan(x + 2 ) then to graph y = cot(x), reflect the graph of y = tan x across the x-axis and shift it left by π2 . π 6 To graph y = cot(x − 2 ), first reflect the graph of y = tan x across the x-axis and shift it left by π2 to obtain the graph of the cotangent function. Finally, shift the graph right by π2 . π 7 To graph y = cot(2x − 2 ), first reflect the graph of y = tan x across the x-axis and shift it left by π2 to obtain the graph of the cotangent function. Then shift the graph right by π4 and then shrink the function by a factor of two in the horizontal direction. Smith (SHSU) 4 To graph y = 5 tan(x − π2 ) + 1, shift the graph of the tangent function right by π2 , stretch it vertically by a factor of 5 and then move the function up 1. Elementary Functions 2013 19 / 22 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 = 225◦ = 5π/4, the angle in the third quadrant with reference angle π/4. But there are many more solutions; if we add 2π to θ, we get new angles that satisfy this equation. Therefore { π 5π + 2πk : k ∈ Z} ∪ { + 2πk : k ∈ Z} 4 4 is the (infinite) set of all solutions. However, recall that the tangent function has period π. So we could simplify this answer by just writing { Smith (SHSU) π + πk : k ∈ Z} 4 Elementary Functions 2013 20 / 22 Two more worked problems 9 Worked problems on graphing trig functions The angle θ has the property that sec θ = 2 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 2 then cos(θ) = 21 . Since the tangent of θ is negative then θ is in the fourth √ quadrant and we may √ π 3 assume θ = −30◦ = − . Then sin(θ) = − and tan(θ) = − 3 3 2 and the other functions are reciprocals of these. Smith (SHSU) Elementary Functions 2013 21 / 22 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) Smith (SHSU) Elementary Functions 2013 22 / 22