Consider the graph of the polar function 𝑟=𝑓(𝜃), where 𝑓(𝜃)=1+2sin𝜃, in the polar coordinate system for 0≤𝜃≤2𝜋. Which of the following statements is true about the distance between the point with polar coordinates (𝑓(𝜃),𝜃) and the origin?ResponsesThe distance is increasing for 0≤𝜃≤𝜋2, because 𝑓(𝜃) is positive and increasing on the interval.The distance is increasing for 0 less or equal than theta less or equal than pi over 2 , because f open parentheses theta close parentheses is positive and increasing on the interval.The distance is increasing for 3𝜋2≤𝜃≤11𝜋6, because 𝑓(𝜃) is negative and increasing on the interval.The distance is increasing for fraction numerator 3 pi over denominator 2 end fraction less or equal than theta less or equal than fraction numerator 11 pi over denominator 6 end fraction , because f open parentheses theta close parentheses is negative and increasing on the interval.The distance is decreasing for 0≤𝜃≤𝜋2, because 𝑓(𝜃) is positive and decreasing on the interval.The distance is decreasing for 0 less or equal than theta less or equal than pi over 2 , because f open parentheses theta close parentheses is positive and decreasing on the interval.The distance is decreasing for 3𝜋2≤𝜃≤11𝜋6, because 𝑓(𝜃) is negative and decreasing on the interval.
Question
Consider the graph of the polar function 𝑟=𝑓(𝜃), where 𝑓(𝜃)=1+2sin𝜃, in the polar coordinate system for 0≤𝜃≤2𝜋. Which of the following statements is true about the distance between the point with polar coordinates (𝑓(𝜃),𝜃) and the origin?ResponsesThe distance is increasing for 0≤𝜃≤𝜋2, because 𝑓(𝜃) is positive and increasing on the interval.The distance is increasing for 0 less or equal than theta less or equal than pi over 2 , because f open parentheses theta close parentheses is positive and increasing on the interval.The distance is increasing for 3𝜋2≤𝜃≤11𝜋6, because 𝑓(𝜃) is negative and increasing on the interval.The distance is increasing for fraction numerator 3 pi over denominator 2 end fraction less or equal than theta less or equal than fraction numerator 11 pi over denominator 6 end fraction , because f open parentheses theta close parentheses is negative and increasing on the interval.The distance is decreasing for 0≤𝜃≤𝜋2, because 𝑓(𝜃) is positive and decreasing on the interval.The distance is decreasing for 0 less or equal than theta less or equal than pi over 2 , because f open parentheses theta close parentheses is positive and decreasing on the interval.The distance is decreasing for 3𝜋2≤𝜃≤11𝜋6, because 𝑓(𝜃) is negative and decreasing on the interval.
Solution
The correct statement is: "The distance is increasing for 0≤𝜃≤𝜋/2, because 𝑓(𝜃) is positive and increasing on the interval."
Here's why:
The polar function 𝑟=𝑓(𝜃)=1+2sin(𝜃) is a sinusoidal function that oscillates between 1 - 2 = -1 and 1 + 2 = 3.
For 0≤𝜃≤𝜋/2 (the first quadrant in the polar coordinate system), the sine function is positive and increasing, which means 𝑓(𝜃) is also positive and increasing.
Since the distance from the origin in polar coordinates is given by the absolute value of the radial coordinate r, and r = 𝑓(𝜃), the distance from the origin is also increasing for 0≤𝜃≤𝜋/2.
The other statements are incorrect because they either refer to intervals where 𝑓(𝜃) is not increasing, or they incorrectly state that 𝑓(𝜃) is negative.
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