Question

What is the range of the sine function?

Short answer

The range of the sine function is \([-1, 1]\).

Step-by-step solution

- Understand the sine function

The sine function is a periodic function defined for all real numbers. It outputs values based on the ratio of the length of the side opposite a given angle to the hypotenuse in a right-angled triangle.

- Analyze the sine wave

Visualize the sine wave: it oscillates smoothly and continually from its maximum value to its minimum value. The sine wave peaks at 1 and dips at -1.

- Identify the maximum and minimum values

From the sine wave, observe that the maximum value is 1 and the minimum value is -1. These are the extreme y-values that the sine function can have.

- State the range

Based on the analysis in the previous steps, the range of the sine function is the set of all y-values it can output, which is from -1 to 1. This can be expressed as \([-1, 1]\).

Key concepts

sine function

The sine function is one of the fundamental trigonometric functions. It is often denoted as \(\text{sin}(x)\). The sine function can be defined for any real number, and it relates to the geometry of a right-angled triangle. Specifically, for a given angle \(x\), the sine of the angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

To visualize the sine function, imagine plotting a point on the unit circle as an angle sweeps from 0 to 360 degrees (or from 0 to \(2\pi\) radians). The \(y\)-coordinate of this point gives us the value of \( \text{sin}(x)\). This process creates a recurring wave-like pattern called a sine wave.

periodic function

A periodic function is a function that repeats its values in regular intervals or periods. The sine function, \(\text{sin}(x)\), is a perfect example of a periodic function.

For the sine function, the period is \(2\pi\) radians. This means that every \(2\backslashpi\) radians, the function values repeat. To see this visually, if you plot the sine function, you'll notice that after moving through one full cycle of \(2\backslashpi\) radians, the wave pattern begins again.

Understanding periodicity is crucial in trigonometry and helps in solving many practical and theoretical problems. Since the sine wave repeats, you can often simplify complex trigonometric problems by considering just one period of the function.

range of function

The range of a function refers to the set of all possible output values (dependent variable). For the sine function, we look at all the values \( \text{sin}(x)\) can take as \(x\) varies over all real numbers.

From the behavior of the sine wave, we can see that it oscillates between a maximum value and a minimum value. By analyzing a complete cycle of the sine wave, it's clear that the highest point it reaches is 1 and the lowest point is -1.

This gives us the range of the sine function: \([ -1, 1 ] \). This means that no matter what angle \(x\) you put into the sine function, the output will always lie between -1 and 1. This characteristic is extremely useful in many areas of mathematics and physics.

trigonometric functions

Trigonometric functions are a group of functions that relate the angles of a triangle to the lengths of its sides. The most commonly used trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

Each of these functions has a specific role in describing circular and oscillatory motion. The sine function, for instance, is used to model wave patterns, such as sound waves, light waves, and tidal waves.

In addition to sine, cosine (which is often written as \(\text{cos}(x)\)) is another fundamental trigonometric function. Cosine gives the ratio of the adjacent side to the hypotenuse of a right-angled triangle. Tangent, written as \(\text{tan}(x)\), is the ratio of the sine of an angle to the cosine of that angle. Understanding these functions and their relationships is crucial in fields such as engineering, physics, and computer science.