Problem 1 - Calculus Single Variable

Question

Calculate the value of the given inverse trigonometric function at the given point. $$ \arcsin (1) $$

Step-by-Step Solution

Answer

The value of \( \arcsin(1) \) is \( \frac{\pi}{2} \).

Step 1: Understand the Definition of Arcsine

The function \( \arcsin(x) \) is the inverse of the sine function, defined for \( -1 \leq x \leq 1 \). It returns the angle \( \theta \) in the interval \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\), where \( \sin(\theta) = x \).

Step 2: Set the Expression

We need to solve \( \arcsin(1) = \theta \) such that \( \sin(\theta) = 1 \).

Step 3: Determine the Angle

Within the interval \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\), the angle \( \theta \) for which \( \sin(\theta) = 1 \) is \( \frac{\pi}{2} \).

Step 4: Conclusion

Thus, the value of \( \arcsin(1) \) is \( \frac{\pi}{2} \).

Key Concepts

Understanding Arcsine

The arcsine function, denoted as \( \arcsin(x) \), plays a key role in trigonometry. As the inverse of the sine function, arcsine "undoes" what the sine function does. Imagine both functions as two sides of a coin: one calculates the ratio of the opposite side to the hypotenuse in a right triangle, while the other deduces the angle from this ratio.
This function is defined only for \(-1 \leq x \leq 1\), which means its input, \(x\), must lie within this range. It returns angles in the interval \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\). If you input \(x = 1\), arcsine will help you find the angle whose sine is 1, staying within this specified range.

The Sine Function Explained

The sine function, often written as \( \sin(\theta) \), is fundamental in trigonometry. It relates to the sides of a right-angled triangle by giving the ratio of the length of the side opposite an angle \( \theta \) to the hypotenuse's length.
Sine values range from \(-1\) to \(1\). At an angle of \( \frac{\pi}{2} \) radians (or 90 degrees), \( \sin(\theta) = 1 \), representing its peak. Thus, when the arcsine function finds \( \arcsin(1) = \frac{\pi}{2} \), it is identifying the angle at which sine achieves its maximum value.
Important aspects of the sine function:

• Periodic: Repeats values every \(2\pi\) radians
• Odd Function: Symmetric around the origin
• Range: Only between \(-1\) and \(1\)

Introduction to Trigonometry

Trigonometry is the field of mathematics that examines the relationships in triangles. Primarily, it looks at right triangles to establish these relationships, using functions like sine, cosine, and tangent.
Key points about trigonometry include:

• Angle Measurement: Often uses radians, where \( \pi \) corresponds to 180 degrees.
• Functions: Involves sine, cosine, and tangent, alongside their inverses like arcsine.
• Applications: Helps in various fields such as physics, engineering, and architecture.

For the problem \( \arcsin(1) \), this function helps determine an angle whose sine is precisely 1. Understanding these fundamental trigonometric concepts is crucial for exploring more complex mathematical ideas and practical applications.