Problem 1. Which numbers are indicated by the following, where n could be any integer?
a) 2nπ
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The even multiples of π:
0, ±2π, ±4π, ±6π, . . .
By '2n' we mean to signify an even number.
b) (2n + 1)π
The odd multiples of π:
±π, ±3π, ±5π, ±7π, . . .
By '2n + 1' we mean to signify an odd number.
Zeros
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 = ?
![](trig_IMG/unit-1.gif)
We saw in Topic 16 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 = nπ.
![](Trig_IMG/113-2.gif)
This will be true, moreover, for any argument of the sine function. For example,
sin 2x = 0 when the argument 2x = nπ;
that is, when
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
![](trig_IMG/138-2.gif)
every 2π units. Hence, sin θ is periodic. Its period is 2π.
Definition. If, for all numbers x, the value of a function at x + p is equal to the value at x --
If f(x + p) = f(x)
-- then we say that the function is periodic and has period p.
![](trig_IMG/sine3-2.gif)
The function y = sin x has period 2π, because
sin (x + 2π) = sin x.
The height of the graph at x is equal to the height at x + 2π -- for all x.
Problem 3.
a) In the function y = sin x, what is its domain?
a) (See Topic 3 of Precalculus.)
x may be any real number. −
< x <
.
b) What is the range of y = sin x?
sin x has a minimum value of −1, and a maximum of +1.
−1
y
1
For example, if a = 2 --
y = sin 2x
-- that means there are 2 periods in an interval of length 2π.
![](trig_IMG/sine4-2.gif)
If a = 3 --
y = sin 3x
-- there are 3 periods in that interval:
![](trig_IMG/sine5-2.gif)
While if a = ½ --
y = sin ½x
-- there is only half a period in that interval:
![](trig_IMG/sine6-2.gif)
The constant a thus signifies how frequently the function oscillates; so many radians per unit of x.
(In physics, when the independent variable is the time t, the constant is written as ω ("omega"). sin ωt. ω is called the angular frequency; so many radians per second.)
Problem 4.
a) For which values of x are the zeros of y = sin mx?
At mx = nπ; that is, at x =
|
nπ m |
. |
b) What is the period of y = sin mx?
|
2π m |
. Since there are m periods in 2π, then one period is 2π |
divided by m. Compare the graphs above.
Problem 5. 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 6. 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 7. y = sin ¼x.
a) What does ¼ indicate?
In an interval of length 2π, there one fourth of a period.
b) What is the period of that function?
2π/¼ = 2π· 4 = 8π
c) Where are its zeros?
![](trig_IMG/tan1-2.gif)
Precalculus.)
Now in the 2nd Quadrant (Fig. 2), the graph has exactly the same negative values (KE') as in the 4th (DE).
![](Trig_IMG/Line3e-2.gif)
And in the 3rd Quadrant (Fig. 3), the graph has exactly the same positive values (KE') as in the 1st (DE).
Thus the graph of Quadrants IV and I is repeated in Quadrants II and III, and periodically along the entire x-axis.
![](trig_IMG/116-2.gif)
This is the graph of y = tan x.
Next Topic: Inverse trigonometric functions
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