18 ## Graphs of the trigonometric functionsLET US BEGIN by introducing some algebraic language. When we write " 0, ±π, ±2π, ±3π, . . .
Problem 1. Which numbers are indicated by the following, where a) 2 To see the answer, pass your mouse over the colored area. The even multiples of π: 0, ±2π, ±4π, ±6π, . . . 2 2 θ and θ + 2 sin θ, therefore, is equal to sin (θ + 2 b) (2 The odd multiples of π: ±π, ±3π, ±5π, ±7π, . . . 2 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? That is, sin θ = 0 when 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, sin θ = 0 when θ = This will be true, moreover, for any argument of the sine function. For example, sin 2 that is, when
Problem 2. Where are the zeros of
At 3
Which numbers are these?
The graph of The zeros of
Here is the graph of The height of the curve at every point is the line value of the sine. In the language of functions,
The independent variable number. We may imagine the unit circle rolled out, in both directions, along the 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. Hence, sin θ 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 function sin ( The height of the graph at Problem 3. a) In the function a) (See Topic 3 of Precalculus.)
b) What is the range of
sin
−1 The graph of The graph of
sum formula can easily prove that. (Topic 20.) 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 4. a) For which values of
b) What is the period of
is 2π divided by
Problem 5. 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 6. 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 7. 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? Consider the line value DE of tan
values. That is, for
− < tan Quadrants IV and I constitute a complete period of
asymptotes. (Topic 18 ofPrecalculus.) Here is the complete graph of The graph of Quadrants IV and I is repeated in Quadrant II (where tan
Problem 8. What is the period of
distance between those two points: π. Next Topic: Inverse trigonometric functions Copyright © 2019 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com Private tutoring available. |