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The sine function, denoted as \( \sin x \), is a fundamental trigonometric function. It describes the ratio of the opposite side to the hypotenuse in a right-angled triangle. The sine function has a periodic nature, repeating its values in regular intervals known as its period, which is \( 2\pi \) for the sine function.
Knowing the sine function is crucial in understanding many mathematical and physical phenomena, including waves and oscillations.

• The range of \( \sin x \) is between \(-1\) and \(1\).
• The function is continuous and smooth, with smooth transitions across its entire range.
• It exhibits symmetry known as odd symmetry, satisfying \( \sin(-x) = -\sin(x) \).
• Critical points within one period are \(x = 0\), \(x = \pi/2\), \(x = \pi\), \(x = 3\pi/2\), and \(x = 2\pi\).

In the exercise, grasping the behavior of the sine function helps us solve for horizontal tangents where the function's rate of change is zero.

The chain rule is a fundamental principle in calculus used for differentiating composite functions. When a function is composed of two or more functions, the chain rule is applied to determine its derivative. The formula for the chain rule involves taking the derivative of the outer function and multiplying it by the derivative of the inner function.
For example, in differentiating \(\sin^2 x\), consider it as \((\sin x)^2\):

• Outer function: \(x^2\)
• Inner function: \(\sin x\)

According to the chain rule, the derivative is: \(2 \sin x \cdot \cos x\).
Applying the chain rule is essential for accurately finding the slope of the tangent line in composite functions, as seen in the exercise.

Differentiation is the process of finding the derivative of a function. The derivative represents the function's rate of change, often interpreted as the slope of the tangent line at any given point on the graph.
In this exercise, differentiation is employed to identify points on the function where the slope is zero, indicating horizontal tangents.
When differentiating \(f(x) = 2\sin x + \sin^2 x\), the individual components \(2\sin x\) and \(\sin^2 x\) are differentiated separately. The basic rules of differentiation and trigonometric identities like the derivative of \(\sin x\), which is \(\cos x\), are applied.

• Derivative of \(2 \sin x\): \(2 \cos x\)
• Derivative of \(\sin^2 x\) utilizing the chain rule: \(2 \sin x \cdot \cos x\)

Differentiation enables us to formulate the equation \(2 \cos x + 2 \sin x \cdot \cos x = 0\), essential for finding solutions.

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent in the context of equations that need solving. Solving these equations often requires knowledge of trigonometric identities and properties.
In the exercise, the trigonometric equation derived from the derivative is \(2 \cos x(1 + \sin x) = 0\). This equation is solved by breaking it down into simpler parts:

• \(2 \cos x = 0\), leading to \(\cos x = 0\) and solutions \(x = \frac{\pi}{2} + n\pi\) (where \(n\) is an integer).
• \(1 + \sin x = 0\), leading to \(\sin x = -1\) and solutions \(x = \frac{3\pi}{2} + 2n\pi\) (where \(n\) is an integer).

Mastering the solution of such equations involves identifying the general solution across all possible values of \(x\), acknowledging its periodicity and symmetry of trigonometric functions.