Introduction to the Sine Function: Properties, Evaluation, and Graphing

f(x)=sin x

To begin, let’s discuss the sine function

To begin, let’s discuss the sine function.

The sine function, denoted as sin(x), is a mathematical function that maps each angle x to the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. However, when we talk about sin(x) as a function, we usually refer to it in terms of its graph, which is a periodic curve that oscillates between -1 and 1.

The sine function has several key properties:

1. Periodicity: The sine function is periodic, meaning it repeats itself after a certain interval. Specifically, sin(x) has a period of 2π radians or 360 degrees. This means that sin(x) = sin(x + 2π) = sin(x + 4π) and so on.

2. Amplitude: The amplitude of the sine function is the absolute value of the maximum value of the function. In this case, the amplitude is 1 because the maximum value of sin(x) is 1 and the minimum value is -1.

3. Symmetry: The sine function is an odd function, which means it has symmetry about the origin. Geometrically, this means that sin(x) = -sin(-x).

4. Zeroes: The sine function has zeroes at x = 0, x = π, x = 2π, and so on. These are the points on the graph where the function crosses the x-axis.

Now, let’s look at the specific function f(x) = sin(x).

For this function, the variable x represents the input to the function, usually an angle measured in radians. The function sin(x) will output the value of the sine of that angle.

To evaluate f(x), you simply substitute the given value of x into the function. For example, if you want to find f(π/6), you would substitute π/6 into the function:

f(π/6) = sin(π/6).

Using the unit circle or trigonometric principles, we can determine that sin(π/6) is equal to 1/2. Therefore, f(π/6) = 1/2.

To graph the function f(x) = sin(x), you can plot points on a set of axes by selecting certain x-values and calculating the corresponding y-values using sin(x). Since sin(x) is a continuous function, you can connect the points to form a smooth curve. The resulting graph will have the shape of a sinusoidal wave, oscillating between -1 and 1.

I hope this explanation helps you understand the sine function and how to evaluate the function f(x) = sin(x). If you have any further questions or need additional clarification, please let me know.

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