Why is arccos(-1/2) Equal to 2π/3? Understanding the Significance of the Range
In this article, we will explore the reasoning behind why arccos(-1/2) is equal to 2π/3. We will discuss the definition of the inverse cosine function, arccos, and why certain angle values fall within or outside its defined range.
Definition of arccos and Its Range
The arccos x function (also known as the inverse cosine function) returns the angle θ whose cosine is x. However, it is important to note that the arccos function is not the same as the reciprocal of the cosine function. The domain of the arccos function is -1 ≤ x ≤ 1, and the range is defined as [0, π]. This means that the output of the arccos function will always be an angle between 0 and π radians (or 0° and 180°).
Why arccos(-1/2) 2π/3?
To understand why arccos(-1/2) 2π/3, consider the cosine function's behavior. The cosine of 2π/3 radians is -1/2. This is a property of the unit circle, where the cosine of 2π/3 corresponds to the x-coordinate of the point on the unit circle at 2π/3 radians. Since -1/2 falls within the range of the arccos function, 2π/3 is a valid output.
However, it is equally valid to say that the cosine of -2π/3 is also -1/2. But why is 2π/3 chosen as the output for arccos(-1/2)? This choice is primarily due to the convention of defining the range of the arccos function as [0, π]. This ensures that the function is single-valued and consistent.
Why -2π/3 Does Not Work As Well
While it is true that the cosine of -2π/3 is also -1/2, -2π/3 is not within the range of the arccos function. The arccos function is not defined for angles outside the interval [0, π]. This is why -2π/3 cannot be the output of the arccos function. The definition of the range ensures that the arccos function is well-defined and bijective (one-to-one and onto).
Periodicity of Trigonometric Functions
Trigonometric functions like cosine are periodic, meaning they repeat their values at regular intervals. For example, the cosine function has a period of 2π. This periodicity implies that the inverse functions, such as arccos, also have multiple values for the same input, except for the conventionally chosen principal value.
For instance, both 2π/3 and -2π/3 (as well as 4π/3, -4π/3, etc.) yield a cosine value of -1/2. However, the arccos function is designed to return the principal value, which is the unique value within the range [0, π]. This is why arccos(-1/2) 2π/3 and not -2π/3.
Choices in Defining the Range
The choice of the range [0, π] for the arccos function is somewhat arbitrary. Other ranges could be chosen, but the convention [0, π] is probably the most reasonable because it covers half the circle, ensuring a one-to-one correspondence between the input and the output. For instance, the range [-π/2, π/2] is used for the arccos function when defined as the inverse sine function, and this choice is also valid and consistent with the one-to-one principle.
Summary
In conclusion, while -2π/3 does yield the correct cosine value, it is not the output of the arccos function due to the restrictions on its range. The arccos function is designed to return a single value within the range [0, π] for a given input, ensuring consistency and well-definedness. Other choices, such as 5π/3, would also be valid in different contexts, but the choice of [0, π] is the most conventional and practical choice for the arccos operation.