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The sine function, denoted as \(\sin \theta\), is one of the fundamental trigonometric functions. It represents the y-coordinate of a point on the unit circle corresponding to an angle \(\theta\). The sine function is periodic and oscillates between -1 and 1. This means for any angle \(\theta\), the result of \(\sin \theta\) will always be within this range. Hence, it repeats itself after every \(2\pi\) radians (360 degrees).

Key characteristics of the sine function include:

• Amplitude: The maximum value of the sine function is 1, and the minimum is -1, giving it an amplitude of 1.
• Period: The sine function completes one full cycle every \(2\pi\) radians.
• Symmetry: The sine function is an odd function, meaning it is symmetric about the origin.

This wave-like oscillation is central to solving problems related to sine, such as finding maxima, minima, and zeros.

Maxima and minima refer to the highest and lowest points a function can reach. For sine functions, they can be easily identified due to the nature of their wave patterns. The maximum point of the sine function \(\sin\theta\) is 1, occurring at angles such as \(\frac{\pi}{2}, \frac{5\pi}{2}\), etc. Conversely, the minimum point, or minima, is -1, occurring at angles such as \(\frac{3\pi}{2}, \frac{7\pi}{2}\), etc.

In the context of our exercise, the modified sine function \(r = 10 - 10\sin\theta\) has been transformed so that it reaches its peak or maximum value when \(\sin\theta\) reaches its minimum, -1. At this point, replacing \(\sin\theta\) with -1, \(r\) achieves its maximum value of 20:
\[ r = 10 - 10(-1) = 20 \]
This transformation shows how altering a base trigonometric function can result in different maxima and minima in its derivative forms.

The zeros of a function are the points where the function equals zero. For the sine function \(\sin\theta\), the zeros appear at angles like \(n\pi\), where \(n\) is an integer, as \(\sin\theta = 0\).

When considering the function \(r = 10 - 10\sin\theta\), the zeros occur when \(r\) equals zero. To find these points, we set the function equal to zero and solve for \(\theta\):

• \(10 - 10\sin\theta = 0\)
• Solve to get \(\sin\theta = 1\).
• The angles satisfying \(\sin\theta = 1\) are: \(\theta = n\pi + (-1)^n \frac{\pi}{2}\), where \(n\) is an integer.

This pattern helps determine all possible values of \(\theta\) where the function \(r\) reaches zero, emphasizing the cyclic nature of trigonometric functions.