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18 Graphs of the trigonometric functionsLET US BEGINwith some algebraic language. When we write "nπ," where n could be any integer, we mean "any multiple of π." 0, ±π, ±2π, ±3π, . . . By the zeros of sin θ we mean those values of θ for which sin θ will equal 0. Now, where are the zeros of sin θ? We saw in Topic 15 on the unit circle that the value of sin θ is equal to the y-coordinate. Hence, sin θ = 0 at θ = 0 and θ = π -- and at all angles coterminal with them. In other words, (1) . . . . . . . . . sin θ = 0 when θ = nπ
They will be the x-intercepts of the graph of the sine. Line (1) will be true, moreover, for any argument θ. For example, sin 2x = 0 when 2x = nπ; that is, when
Problem 1. Where are the zeros of y =sin 3x? At 3x = nπ; that is, at
Which numbers are these?
The period of a function When the values of a function regularly repeat themselves, we say that the function is periodic. The values of sin θ regularly repeat themselves every 2π units. sin (θ + 2π) = sin θ. sin θ therefore is periodic. Its period is 2π. (See the previous topic, Line values.) Definition. If, for all values of x, the value of a function at x + p If f(x + p) = f(x) -- then we say that the function is periodic and has period p. The graph of y = sin x The zeros of y = sin x are at the multiples of π. It is there that the graph crosses the x-axis, because there y = 0. And its period is 2π. But what is the maximum value of the graph and what is its minimum value? sin x has a maximum value of 1 at Here is the graph of y = sin x: The independent variable x is the radian measure. x may be any real number. We may imagine the unit circle rolled out, in both directions, along the x-axis. (See Topic 15: The unit circle.) Problem 2. Vocabulary. a) In the function y = sin x, what is its domain? − b) What is the range of y = sin x? −1 ≤ sin x ≤ 1. The graph of y = cos x The graph of y = cos x is the graph of y = sin x translated On the other hand, it is possible to see directly that Topic 16. Angle CBD is a right angle. The graph of y = sin ax Since the graph of y = sin x has period 2π, then the constant a in y = sin ax indicates the number of periods in an interval of length 2π. (In y = sin x, a = 1.) For example, if a = 2 -- y = sin 2x -- that means there are 2 periods in an interval of length 2π. If a = 3 -- y = sin 3x -- there are 3 periods in that interval: While if a = ½ -- y = sin ½x -- there is only half a period in that interval: The constant a thus signifies how frequently the function oscillates; so many radians per unit of x. (When the independent variable is the time t, as it often is in physics, then the constant is written as ω ("omega"): sin ωt. ω is called the angular frequency; so many radians per second.) Problem 3. a) For which values of x are the zeros of y = sin mx?
b) What is the period of y = sin mx?
is 2π divided by m. Compare the graphs above. Problem 4. y = sin 2x. a) What does the 2 indicate? In an interval of length 2π, there are 2 periods. b) What is the period of that function?
c) Where are its zeros?
Problem 5. y = sin 6x. a) What does the 6 indicate? In an interval of length 2π, there are 6 periods. b) What is the period of that function?
c) Where are its zeros?
Problem 6. y = sin ¼x. a) What does ¼ indicate? In an interval of length 2π, there is one fourth of a period. b) What is the period of that function? 2π/¼ = 2π· 4 = 8π. c) Where are its zeros?
The graph of y = tan x Here is one period of the graph of y = tan x: Why is that the graph? It has effectively been explained in the previous topic, where we considered the line value DE of tan x in quadrants IV and I. In quadrant IV, tan x -- the line value DE -- is negative and takes on all possible negative values: −∞ < tan x < 0. At x = 0, tan x = 0. And finally in quadrant I, tan x takes on all positive values: 0 < tan x < ∞. And so in the interval from − Here again is the graph. At the quadrantal angles − Here is the complete graph of y = tan x. The graph of Quadrants IV and I is repeated in Quadrant II (where tan x is negative) and quadrant III (where tan x is positive), and periodically along the entire x-axis. Problem 7. What is the period of y = tan x?
distance between those two points: π. Next Topic: Inverse trigonometric functions Copyright © 2022 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |