18 ## Graphs of the trigonometric functionsLET US BEGINwith some algebraic language. When we write " 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 =
They will be the Line (1) will be true, moreover, for any argument θ. For example, sin 2 that is, when
Problem 1. Where are the zeros of
At 3
Which numbers are these?
The 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.)
If -- then we say that the function is periodic and has period The graph of The zeros of sin Here is the graph of The independent variable We may imagine the unit circle rolled out, in both directions, along the Problem 2. Vocabulary. a) In the function − < b) What is the range of
−1 ≤ sin The graph of The graph of On the other hand, it is possible to see directly that Topic 16. Angle CBD is a right angle. The graph of Since the graph of
indicates the number of periods in an interval of length 2π. (In For example, if
-- that means there are 2 periods in an interval of length 2π. If
-- there are 3 periods in that interval: While if
-- there is only half a period in that interval: The constant (When the independent variable is the time Problem 3. a) For which values of
b) What is the period of
is 2π divided by Compare the graphs above.
Problem 4. 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. 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. 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 c) Where are its zeros?
The graph of Here is one period of the graph of Why is that the graph? It has effectively been explained in the previous topic, where we considered the line value DE of tan In quadrant IV, tan −∞ < tan At And finally in quadrant I, tan 0 < tan And so in the interval from − to , tan Here again is the graph. At the quadrantal angles − and , tan Here is the complete graph of The graph of Quadrants IV and I is repeated in Quadrant II (where tan
Problem 7. What is the period of
distance between those two points: π. Next Topic: Inverse trigonometric functions Copyright © 2022 Lawrence Spector Questions or comments? E-mail: [email protected] |