Understanding Cosec A Given Sin-1 (1/3)

In trigonometry, understanding the relationships between different trigonometric functions is crucial. This article explores how to determine the value of cosec A given that sin?1 (1/3) A. This relationship can be explained through a series of direct substitutions and reciprocal functions.

Introduction to Trigonometric Functions

The functions sine (sin) and cosecant (cosec) are both fundamental in trigonometry. The cosecant function, cosec A, is the reciprocal of the sine function, sin A. In other words, cosec A 1 / sin A.

Given Information and Substitution

We are given the information that sin?1 (1/3) A. This implies that the angle A is such that the sine of A is equal to 1/3, i.e., sin A 1/3.

Direct Substitution

Since we know that cosec A 1 / sin A, we can directly substitute sin A with 1/3.

cosec A 1 / (1/3) 3

Simplifying the Reciprocal Function

Reciprocating the sine function is a straightforward process. If we have sin A 1/3, then taking the reciprocal of both sides, we get:

cosec A 1 / (1/3) 3

Understanding the Reciprocal Relationship

The relationship between cosec A and sin A is a reciprocal one. If sin A 1/Cosec A, then when we are given sin A 1/3, it directly means that Cosec A 3.

Verification

We can verify this by checking the reciprocal relationship:

Given sin A 1/3, we substitute into the reciprocal formula:

Cosec A 1 / (1/3) 3

Conclusion

In summary, the value of cosec A when sin?1 (1/3) A is 3. This relationship is underpinned by the basic definition of the trigonometric functions and their reciprocals. Understanding these relationships is essential for solving more complex trigonometric problems and is a fundamental skill in trigonometry.