Understanding the Equation
To find sin alpha from the given equation, we start by rewriting the expression for tan alpha:
- tan alpha = (sin theta - cos theta) / (sin theta cos theta)
This implies that:
- tan alpha = (A) / (B) where A = sin theta - cos theta and B = sin theta cos theta.
Using the Identity for Sin Alpha
Now, we know that:
- sin alpha = tan alpha / sqrt(1 + tan^2 alpha)
To find sin alpha, we first need to compute tan^2 alpha:
- tan^2 alpha = [(sin theta - cos theta) / (sin theta cos theta)]^2
This can be simplified as:
- tan^2 alpha = (sin^2 theta - 2sin theta cos theta + cos^2 theta) / (sin^2 theta cos^2 theta)
Since sin^2 theta + cos^2 theta = 1:
- tan^2 alpha = (1 - 2 sin theta cos theta) / (sin^2 theta cos^2 theta)
Finding Sin Alpha
Now, substitute tan alpha and tan^2 alpha in the sin alpha formula:
- sin alpha = [(sin theta - cos theta) / (sin theta cos theta)] / sqrt[1 + (1 - 2 sin theta cos theta) / (sin^2 theta cos^2 theta)]
After simplification, we will get the expression for sin alpha.
Final Expression
Thus, the final expression for sin alpha can be derived as:
- sin alpha = (sin theta - cos theta) / sqrt[(sin^2 theta + cos^2 theta)(sin^2 theta cos^2 theta)]
This ultimately leads to the required value of sin alpha based on the values of sin theta and cos theta.
Conclusion
To summarize, by understanding the relationship between tan alpha and its components, we can effectively derive sin alpha.