Identify the circular function that satisfies each description...

Question

Identify the circular function that satisfies each description.

period is π; function is decreasing on the interval (0, π)

Accepted Answer

Welcome back. Everyone. In this problem, we want to write the function that has the properties of increasing on the interval between negative half of Pi and half of Pi and having a period equal to pi. For our answer choices. A says that our function is the tangent of XB says it is the core tangent of XC says it's the sine of X and D says it's the cosine of X. Now, let's think about each of these properties and try to think about which trigonometric functions have that property. Let's start with the fact that it's increasing on increasing on the interval between half of pi negative half of pi sorry and half of pi. Now, what does that look like? But if we were to sketch a graph here, right? OK. And a graph of pi against our function Y OK. And we're going from negative half of pie to pay, then if we, if we think about it, there are two functions that fit that criteria. The first that we can think of, oops let me put that properly here. The first that we can think of is the tangent function. OK? Because the tangent function looks something like this. OK. And the other one that we can think of is the sine function because the sine function looks something like this as well. OK. So let me see if I can have that nicely drawn here because remember that it, it looks somewhat like this can it, and it goes along. OK. So both of these are increasing, but now which one of them are we looking for? So this is the sine function and in red, it's the tangent. No, to figure that out, let's think about the second criteria. It says that it has a period equal to pi. So which one of those functions is it? Well, the tangent has a period equal to pi. So the period of the tangent equals pi if we can recall. OK. So let's write that down. OK. And on the other hand, the period of the sine function period of sine is actually equal to two pi. How do we know that? Well, recall that for the tangent right, the tangent to find the period of the tangent, it would be pi divided by the coefficient of in this case X which is one and pi divided by one equals one. On the other hand, the period of the sine function would be two pi divided by the coefficient of X which in this case is one and two pi divided by one equals two pi. So that's how we know for sure that Y is the correct answer. Therefore, the tangent function is increasing on the interval between negative half of pi and half of pi and having a period equal to pi. Thanks a lot for watching everyone. I hope this video helped.

Identify the circular function that satisfies each description.
period is π; function is decreasing on the interval (0, π)

Key Concepts

Period of a Trigonometric Function

Monotonicity of Functions on an Interval

Common Circular Functions and Their Properties

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