Equivalent Of Arcsin(tan(π/4)) In Radians

In the realm of trigonometry, understanding inverse trigonometric functions is crucial for solving various mathematical problems. One such function is the arcsine, denoted as $\sin^{-1}(x)$, which gives the angle whose sine is x. This article delves into finding the equivalent of the expression $\sin^{-1}(\tan(\frac{\pi}{4}))$ in radians. We will explore the fundamental concepts of trigonometric functions, their inverses, and how to evaluate such expressions step by step. By the end of this exploration, you'll have a clear understanding of the process and the correct answer.

To effectively tackle the given expression, let's first establish a solid foundation in trigonometric functions and their inverses. Trigonometric functions, namely sine, cosine, tangent, cotangent, secant, and cosecant, relate the angles of a right triangle to the ratios of its sides. The sine function, denoted as sin(x), gives the ratio of the opposite side to the hypotenuse in a right triangle. The cosine function, cos(x), gives the ratio of the adjacent side to the hypotenuse, and the tangent function, tan(x), gives the ratio of the opposite side to the adjacent side.

Inverse trigonometric functions, on the other hand, perform the reverse operation. They take a ratio as input and return the angle that corresponds to that ratio. The inverse sine function, arcsine or $\sin^{-1}(x)$, gives the angle whose sine is x. Similarly, the inverse cosine function, arccosine or $\cos^{-1}(x)$, gives the angle whose cosine is x, and the inverse tangent function, arctangent or $\tan^{-1}(x)$, gives the angle whose tangent is x. It's important to note that inverse trigonometric functions have restricted ranges to ensure they are well-defined. For instance, the range of arcsine is $[-\frac{\pi}{2}, \frac{\pi}{2}]$, the range of arccosine is $[0, \pi]$, and the range of arctangent is $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Understanding these concepts is paramount for simplifying and evaluating trigonometric expressions involving inverse functions. Now, let's proceed to solve the given problem by breaking it down into manageable steps.

Now, let's embark on a step-by-step journey to evaluate the expression $\sin^{-1}(\tan(\frac{\pi}{4}))$ and unveil its equivalent in radians. This process involves understanding the order of operations and applying our knowledge of trigonometric functions.

Step 1: Evaluate the Inner Function: tan(π/4)

The first step in simplifying the expression is to evaluate the inner function, which is $\tan(\frac{\pi}{4})$ The tangent function, as we discussed earlier, is the ratio of the opposite side to the adjacent side in a right triangle. The angle $\frac{\pi}{4}$ radians, which is equivalent to 45 degrees, is a special angle in trigonometry. We know that in a right triangle with angles 45-45-90, the opposite and adjacent sides are equal in length.

Therefore, $\tan(\frac{\pi}{4}) = \frac{\sin(\frac{\pi}{4})}{\cos(\frac{\pi}{4})} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$. This means that the tangent of $\frac{\pi}{4}$ radians is equal to 1. We have successfully simplified the inner function, and our expression now becomes $\sin^{-1}(1)$.

Step 2: Evaluate the Outer Function: arcsin(1)

With the inner function evaluated, we now turn our attention to the outer function, which is $\sin^{-1}(1)$. Recall that the arcsine function, $\sin^{-1}(x)$, gives the angle whose sine is x. In this case, we need to find the angle whose sine is 1. We are looking for an angle $\ heta$ such that $\sin(\ heta) = 1$.

Considering the range of the arcsine function, which is $[-\frac{\pi}{2}, \frac{\pi}{2}]$, we need to find an angle within this range whose sine is 1. We know that the sine function represents the y-coordinate on the unit circle. The y-coordinate is 1 at the angle $\frac{\pi}{2}$ radians (90 degrees). Therefore, $\sin^{-1}(1) = \frac{\pi}{2}$.

Thus, the equivalent of $\sin^{-1}(\tan(\frac{\pi}{4}))$ in radians is $\frac{\pi}{2}$. This completes our step-by-step evaluation, leading us to the final answer.

In this comprehensive exploration, we have successfully determined the equivalent of the expression $\sin^{-1}(\tan(\frac{\pi}{4}))$ in radians. By breaking down the problem into manageable steps, we first evaluated the inner function, $\tan(\frac{\pi}{4})$, which simplified to 1. Then, we evaluated the outer function, $\sin^{-1}(1)$, which gave us the final answer of $\frac{\pi}{2}$ radians.

This exercise highlights the importance of understanding trigonometric functions and their inverses. A solid grasp of these concepts is essential for tackling more complex mathematical problems in trigonometry and calculus. By mastering the step-by-step approach and familiarizing ourselves with special angles and their trigonometric values, we can confidently navigate through such evaluations. Remember, practice and consistent effort are key to building proficiency in mathematics. This journey into the world of trigonometric functions and their inverses has not only provided us with a solution to a specific problem but has also strengthened our understanding of the fundamental principles that govern this fascinating branch of mathematics.

The equivalent of $\sin^{-1}(\tan(\frac{\pi}{4}))$ in radians is B. $\frac{\pi}{2}$.