True or False. Where the rate of change of y=sin, x is positive. And when the rate of change y=cos x is negative.
Explain how you know
You want to try that again in English?
Sorry, English is no my first language.
That was the wording of the question I was given, I was very confused reading it.
The statement "Where the rate of change of y=sin x is positive" is false.
To understand why, we need to consider the derivatives of the functions sine and cosine. The derivative of the sine function is the cosine function, and the derivative of the cosine function is the negative sine function.
In mathematical terms, if we have the function y = sin(x), then the rate of change (the derivative) of y with respect to x is dy/dx = cos(x).
So, when we evaluate the derivative at any given x value, the sign of the derivative depends on the value of cosine at that particular x.
To determine whether the rate of change is positive or negative, we need to look at the sign of the cosine function.
The cosine function assumes positive values in the first and fourth quadrants of the unit circle (i.e., when x is between 0 and 2π, or when x is between -2π and 0). In those quadrants, the derivative (rate of change) of the sine function is positive. However, in the second and third quadrants, the cosine function takes on negative values, so the derivative of the sine function is negative in those parts of the unit circle.
Therefore, the correct statement would be: "Where the rate of change of y = sin(x) is positive for x in the first and fourth quadrants of the unit circle."
Similarly, for the statement "When the rate of change of y = cos(x) is negative," we can infer that it is true. The derivative of the cosine function is -sin(x), which means that the rate of change is negative whenever the sine function is negative.
To summarize, the rate of change (derivative) of y = sin(x) is positive when x is in the first and fourth quadrants of the unit circle, and the rate of change (derivative) of y = cos(x) is negative when x is in the second and third quadrants of the unit circle.