Is y equal to cotx, tanx, cosx, or sinx if its graph is periodic with a period of 2pi, an amplitude equal to 1, and it passes through the origin?

Plot the graphs. It is equal to sin(x).

Proof: cot(x), tan(x) have period pi and no amplitude in the classical way.
cos(0)=1, hence it cannot pass through the origin, whereas sin(0) = 0, yielding sin(x) to be the function-in-question.

thanks.

To determine whether y is equal to cot(x), tan(x), cos(x), or sin(x) based on its characteristics, we need to analyze the given information. Let's break it down:

1. Period of 2π: The period of a function refers to the length of one complete cycle. In this case, since the period is 2π, it means that the graph repeats itself every 2π units along the x-axis.

2. Amplitude of 1: The amplitude determines the maximum value of the graph's oscillation. Since the amplitude is 1, it indicates that the absolute value of the function's range is restricted to the interval [0, 1].

3. Passes through the origin: This means that the y-intercept of the graph is at (0, 0). In other words, when x = 0, y = 0.

Given these characteristics, we can eliminate cot(x) as a possible option since its graph is not periodic with a period of 2π and does not pass through the origin.

Now, let's consider the remaining three options:

1. tan(x): The tan(x) function, or tangent function, has a period of π and it is not aligned with the origin. Therefore, it does not meet the criteria specified.

2. cos(x): The cos(x) function, or cosine function, has a period of 2π, which matches the given information about the graph's period. However, cosine functions do not pass through the origin, so cos(x) is also not the correct answer.

3. sin(x): The sin(x) function, or sine function, has a period of 2π, just like the given period of the graph. Additionally, it has an amplitude of 1, which matches the given amplitude. Moreover, since sin(0) = 0, it passes through the origin. Therefore, y is likely equal to sin(x) based on the provided information.

In conclusion, given that the graph is periodic with a period of 2π, has an amplitude of 1, and passes through the origin, y is most likely equal to sin(x).