Try to answer the questions by referring to the figure, without looking back in the lext or usin

Name: Problem 7, Chapter 5: Precalculus: Graphs and Models
Uploaded: 2020-06-26T19:33:30-08:00
Duration: 7 min 27 s
Channel: Peter Winans
Description: Try to answer the questions by referring to the figure, without looking back in the lext or using a calculator.(FIGURE CAN'T COPY) What are the periods of the sine, cotangent, and cosecant functions?

Try to answer the questions by referring to the figure,...

Key Concepts

Cosecant Function

The cosecant function is the reciprocal of the sine function. Because it is directly related to the sine function, it shares the same period of 2?, repeating its values every 2? units along the horizontal axis.

Cotangent Function

The cotangent function, defined as the ratio of the cosine function to the sine function, has a period of ?. This shorter cycle compared to the sine function indicates that its values repeat every ? units.

Periodicity

Periodicity in mathematics refers to the property of a function to repeat its values in regular intervals or periods. This concept is fundamental when analyzing trigonometric functions because each function has a characteristic interval after which its behavior repeats.

Sine Function

The sine function is a primary trigonometric function that exhibits periodic behavior. Its repeating interval, called the period, is 2?, meaning that after every 2? units along the horizontal axis, the function repeats its cycle.

Transcript

00:03 All right, we are going to be using the unit circle to discuss the period for the sign function, the contangent function, and the co -secant functions.

00:29 Those are the three functions that we're going to be talking about right now.

00:35 So if we talk about the period, we're talking about the length of time that it takes.

00:40 These are all periodic functions.

00:42 And they're periodic functions because they repeat a pattern.

00:46 So we're talking about the length of time it takes before we start repeating the same pattern.

00:53 So when we talk about the sign function, if we kind of unwrap the unit circle, so to speak, we have on the unit circle at zero, we know on the unit circle sign is equal to the y coordinate.

01:10 And then at pi over two, it's equal to one.

01:13 At pi we're at 0, 3 pi over 2 we're at negative 1.

01:17 So if we kind of take those values, pi over 2, pi, 3 pi over 2, and then 2 pi, then this graph would look like this.

01:40 At 0, y is 0, at pi over 2, y is equal to 1.

01:50 At pi, sine is equal to 0 again, at 3 pi over 2, sign is equal to negative 1, 2 pi were back to 0.

02:02 Now, what's going to happen? if i go around the circle again, right? 5 pi over 2 would be next.

02:11 Then i'd be back up to 1.

02:13 So, in other words, our sine curve is doing something like this.

02:17 And then now, if we go around the circle again, we're just going to repeat that same pattern.

02:24 So that point right there marks one period of the sine curve.

02:29 In other words, one trip around the unit circle.

02:32 So the period for y equals the sine of x is 2 pi.

02:47 And when we talk about the co -secant function, co -secent and sign are related.

02:52 Co -secent is the reciprocal of the sign function.

02:55 So the thing i want to point out is that sine is equal to 0 at 0.

03:03 Well, if i take the reciprocal of 0, you get 1 over 0.

03:06 That means the cosecant function is going to be undefined where sign is equal to zero.

03:12 So it's going to be undefined at 0, at pi, and at 2 pi.

03:19 The reciprocal of 1 is 1...

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