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 = ?

Problem 2.   Where are the zeros of  y = sin 3x?

At 3x = nπ; that is, at

Which numbers are these?

The graph of y = sin x

The zeros of y = sin x are at the multiples of π.  And it is there that the graph crosses the x-axis.  But what is the maximum value of the graph, and what is its minimum value?

Here is the graph of y = sin x:

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.

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.

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?

The graph of y = cos x

The graph of y = sin ax

When a function has this form,

y = sin ax,

then the constant a indicates the number of periods in an interval of length 2π.

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.

(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.

The graph of y = tan x

Precalculus.)

Now in the 2nd Quadrant (Fig. 2), the graph has exactly the same negative values (KE') as in the 4th (DE).

And in the 3rd Quadrant (Fig. 3), the graph has exactly the same positive values (KE') as in the 1st (DE).