Properties of Sine Function - Algebra-2

1. Fundamental Concepts

Definition: The sine function, denoted as $$\sin(x)$$ , is a periodic function with a period of $$2\pi$$ . It represents the y-coordinate of a point on the unit circle corresponding to an angle $$x$$ .
Domain and Range: The domain of $$\sin(x)$$ is all real numbers ( $$\mathbb{R}$$ ), and its range is $$[-1, 1]$$ .
Odd Function: The sine function is odd, meaning $$\sin(-x) = -\sin(x)$$ .

2. Key Concepts

Amplitude: The amplitude of the basic sine function $y = \sin x$ is $1$ (amplitude is half the difference between the maximum and minimum values of the function, i.e., $\frac{1 - (-1)}{2} = 1$), which determines the vertical fluctuation range of the function graph.

Zeros: When $x = k\pi$ ($k \in \mathbb{Z}$), $\sin x = 0$, so the intersection points of the function graph with the $x$-axis are $(k\pi, 0)$, such as $(0, 0)$, $(\pi, 0)$, $(2\pi, 0)$, etc.

Maximum and Minimum Points:

Maximum Value: When $x = \frac{\pi}{2} + 2k\pi$ ($k \in \mathbb{Z}$), $\sin x = 1$, and the corresponding highest point on the graph is $(\frac{\pi}{2} + 2k\pi, 1)$.

Minimum Value: When $x = \frac{3\pi}{2} + 2k\pi$ ($k \in \mathbb{Z}$), $\sin x = -1$, and the corresponding lowest point on the graph is $(\frac{3\pi}{2} + 2k\pi, -1)$.

Parity: It is an odd function, satisfying $\sin(-x) = -\sin x$, and its graph is symmetric about the origin of the coordinate system.

3. Examples

(1) Easy Difficulty

Find the function values of $y = \sin x$ at $x = \frac{\pi}{2}$ and $x = \pi$, and indicate the positions of the corresponding points on the graph.

Solution: When $x = \frac{\pi}{2}$, $y = \sin\frac{\pi}{2} = 1$, corresponding to the highest point $(\frac{\pi}{2}, 1)$ on the graph; when $x = \pi$, $y = \sin\pi = 0$, corresponding to the intersection point $(\pi, 0)$ of the graph with the $x$-axis.

(2) Medium Difficulty

Determine the amplitude, smallest positive period of the function $y = 2\sin x$, and write its maximum and minimum values.

Solution: Compared with the basic sine function $y = \sin x$, the amplitude of $y = 2\sin x$ is $|2| = 2$; the period is determined by the coefficient of $x$, which is $1$ here, so the smallest positive period is still $2\pi$; the maximum value is $2\times1 = 2$, and the minimum value is $2\times(-1) = -2$.

(3) Hard Difficulty

It is known that the graph of a sine function passes through the points $(\frac{\pi}{3}, \frac{\sqrt{3}}{2})$ and $(\frac{2\pi}{3}, \frac{\sqrt{3}}{2})$, and its smallest positive period is $2\pi$. Find the expression of this function (Hint: Let the function be $y = A\sin x + B$, $A \neq 0$).

Solution: Substitute the two points into the expression:

1. When $x = \frac{\pi}{3}$, $\frac{\sqrt{3}}{2} = A\sin\frac{\pi}{3} + B = A\times\frac{\sqrt{3}}{2} + B$;

2. When $x = \frac{2\pi}{3}$, $\frac{\sqrt{3}}{2} = A\sin\frac{2\pi}{3} + B = A\times\frac{\sqrt{3}}{2} + B$.

Verify with specific values: Let $B = 0$, then $A = 1$, and the function $y = \sin x$ satisfies the conditions (Note: This is a basic case. If $B \neq 0$, additional conditions are required to determine it. Combined with the period and the two points in this problem, the basic solution is $y = \sin x$).

4. Problem-Solving Techniques

Graphical Interpretation: Use the unit circle to visualize sine values and their periodicity.
Formula Application: Apply the sine function's properties, such as periodicity and amplitude, to solve problems.
Practice with Values: Memorize sine values for common angles to quickly solve problems without a calculator.