Determining the Quadrant for Trigonometric Inverses: sin^-1(4/5) and cos^-1(-4/5)

Trigonometric inverses can be a useful tool in solving problems involving angles and their relationships to sides of triangles. In this article, we explore the quadrants for the specific trigonometric inverses of sin^-1(4/5) and cos^-1(-4/5). We will explain the reasoning behind each determination and provide context for these functions.

Understanding the Ranges of Inverse Trigonometric Functions

First, let's understand the ranges of the inverse trigonometric functions. The arccos function, which is another term for cos^-1, has a range from 0 to π. This means that the output values for the arccos function are restricted to the first and second quadrants.

The arcsin function, which is another term for sin^-1, has a range from -π/2 to π/2. This implies that the arcsin function's output values are restricted to the first and fourth quadrants.

Applying the Ranges to cos^-1(-4/5)

Given the argument -4/5 for cos^-1, we need to determine which quadrant(s) the angle could fall into. Since -4/5 is negative, the angle must lie in the second quadrant, as the range for arccos is [0, π].

To summarize, since arccos x is in the range [0, π], and -4/5 is negative, the angle corresponding to cos^-1(-4/5) must be in the second quadrant.

Considering the Quadrants for sin^-1(4/5)

For the sin^-1(4/5) or arcsin(4/5), we need to determine the quadrant(s) where the sine function yields a positive value. Since 4/5 is positive and the range of arcsin is [-π/2, π/2], the positive sine value corresponds to the first quadrant. Therefore, sin^-1(4/5) must be in the first quadrant.

The reasoning here is that the sine function takes a positive value in the first quadrant, and since 4/5 is positive, the corresponding angle is located in the first quadrant, satisfying the constraint -π/2 ≤ y ≤ π/2.

Conclusion and Summary

In summary, we have determined the following:

The angle corresponding to cos^-1(-4/5) is in the second quadrant. The angle corresponding to sin^-1(4/5) is in the first quadrant.

Understanding the ranges and properties of these inverse trigonometric functions is crucial for solving more complex trigonometric problems. Whether you need to determine angles, solve equations, or work with trigonometric identities, this knowledge provides a solid foundation.