Simplifying the Inverse Sine of a Complex Fraction
In this article, we will explore the process of simplifying an expression involving the inverse sine of a complex fraction. The expression in question is sin-1(√ x / (1x)). By breaking down the components and applying trigonometric identities, we can simplify this expression significantly. Let's delve into the step-by-step process.
Step-by-Step Simplification Process
Let's start by defining:
x tan2t
Using this substitution, we aim to simplify the given expression step by step:
(x / 1x) (tan2t) / (1tan2t) (tan2t) / (sec2t)
Recall the Pythagorean identity for trigonometric functions:
sec2t 1 tan2t
Using this identity, we can rewrite the expression inside the square root:
(tan2t) / (sec2t) (tan2t) / (1 tan2t)
Since x tan2t, we have:
(x) / (1 x)
Now, let's simplify the square root of this expression:
sqrt((x) / (1 x))
This can be written as:
sqrt(x) * sqrt((1 / (1 x)))
Using the trigonometric identity, we know that:
sqrt((1 / (1 x))) sin t
Therefore, we can conclude:
sqrt((x) / (1 x)) sin t
Finally, we take the inverse sine of both sides:
sin-1(sqrt((x) / (1 x))) sin-1(sin t) t
Since x tan2t, we can express t as:
t tan-1(sqrt(x))
Conclusion
By applying trigonometric identities and simplifying the expression step-by-step, we have derived that:
sin-1(sqrt(x / (1x))) tan-1(sqrt(x))
This simplification is particularly useful in calculus, differential equations, and various applications in physics and engineering. Understanding these concepts can greatly enhance your problem-solving skills in mathematics and related fields.