Understanding the Expression cos-1(2x) - 1 and Its Implications

When dealing with mathematical expressions involving the inverse cosine function, it is essential to understand the domain and range of the function. The expression cos-1(2x) - 1 requires a detailed analysis of its components to determine its validity and range of applicability. This article explores the nuances of this trigonometric expression, providing a comprehensive explanation of its behavior and implications.

Domain and Range of the Inverse Cosine Function

The inverse cosine function, denoted as cos-1x, is also known as the arccosine function. It is defined on the domain [-1, 1] and has a range of [0, π] radians (or [0°, 180°]). Therefore, for the expression cos-1(2x) - 1, the value of 2x - 1 must lie within the interval [-1, 1]. This leads to the following inequalities:

-1 ≤ 2x - 1 ≤ 1

Solving these inequalities gives us the range of x:

From 2x - 1 ≥ -1, we get 2x ≥ 0 or x ≥ 0. From 2x - 1 ≤ 1, we get 2x ≤ 2 or x ≤ 1.

Combining these inequalities, we find that:

0 ≤ x ≤ 1

If x is within this interval, the expression cos-1(2x) - 1 is defined and gives an angle corresponding to the cosine value of 2x - 1.

Trigonometric Identities and Simplifications

Another important aspect is the interpretation of the expression based on trigonometric identities. The identity cosab cosacosb - sinasinb can be used to simplify certain expressions. However, in the context of cos-1(2x) - 1, this identity does not directly apply. An alternative approach involves substituting x with a trigonometric function as shown below:

Substitution Approach

By letting x cos2A, the expression cos-1(2x) - 1 can be simplified as follows:

Substituting x cos2A, we get: cos-1(2cos2A - 1) - 1 Using the double-angle identity cos2A 2cos2A - 1, this further simplifies to: cos-1(cos2A) - 1 Finally, applying the property of the inverse cosine function, we obtain: 2A - 1

By setting A cos-1√x, we arrive at the answer:

cos-1(2x) - 1 2cos-1√x - 1

Conclusion

In summary, the expression cos-1(2x) - 1 is defined for x in the interval [0, 1] and represents the angle corresponding to the cosine value of 2x - 1. Understanding the domain and range of the inverse cosine function, as well as utilizing trigonometric identities, is crucial for working with such expressions. This article provided a detailed exploration of the expression and its implications.