Problem 210 - Calculus Volume 1

Question

For the following exercises, evaluate the functions. Give the exact value. $$ \sin ^{-1}(-1) $$

Step-by-Step Solution

Answer

\(\sin^{-1}(-1) = -\frac{\pi}{2}\)

Step 1: Understanding the Problem

We are asked to evaluate the inverse sine function, or arcsine, at the point -1: \(\sin^{-1}(-1)\). The inverse sine function returns an angle whose sine is the given value.

Step 2: Range of Arcsine Function

The range of the \(\sin^{-1}(x)\) function is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) (or -90° to 90°). This means the output will be an angle within this range.

Step 3: Sin Value of -1

Since \(\sin^{-1}(x)\) returns an angle \(\theta\) such that \(\sin(\theta) = x\), we need to find the angle \(\theta\) for which \(\sin(\theta) = -1\).

Step 4: Finding the Angle

The angle \(\theta\) for which \(\sin(\theta) = -1\) within the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) is \(-\frac{\pi}{2}\). Therefore, \(\sin^{-1}(-1) = -\frac{\pi}{2}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsine, serve as the opposites of standard trigonometric functions. They help us find the angles when we know the trigonometric ratio. In simpler terms, if a trigonometric function gives you a number for a certain angle, its inverse helps you find that angle back from the number. For example, the arcsine function is the inverse of the sine function.
To understand how they work, note these important points:

• The input to an inverse trigonometric function is a value that a regular trigonometric function produces. For arcsine, these inputs range from -1 to 1, because a sine value cannot be outside this range.
• The output is an angle, typically given in radians, that corresponds to those inputs.

This function is crucial for solving problems that involve determining angles from given trigonometric values.

Sine Function

The sine function is a fundamental trigonometric function that is often represented as \sin(\theta)\, where \(\theta\) is the angle. It measures the ratio of the opposite side to the hypotenuse in a right-angled triangle.

• The sine of an angle gives us a number, which can vary from -1 to 1. These values reflect the position of a point on the unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane.
• For some specific angles, like \(-\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi \,and \frac{3\pi}{2}\), the sine values are well-known and easy to remember.

When using the arcsine function in problem-solving, it is crucial to understand the relationship between an angle and its sine value. This understanding will help in effectively reversing the process of finding the sine of an angle, associating it back to the angle itself.

Angles in Radians

Radians are a unit of angular measure used widely in mathematics. Unlike degrees, radian measure connects more directly to the radius of a circle. In fact, when you measure angles in radians, you're expressing the distance around a circle's arc as a proportion of the circle's radius.
Here's how to think about it:

• The whole circle is \(2\pi\) radians, similar to how it's 360 degrees.
• Each angle describes a part of the circumference of the unit circle, with one radian equal to the angle created when the arc length equals the radius.

Inversely static functions, like arcsine, often output angles in radians because it simplifies many mathematical calculations. Knowing how to convert between radians and degrees can be helpful. Remember, \(\pi\) radians is equivalent to 180 degrees. Therefore, small segments like \(-\frac{\pi}{2}\) easily translate to typical angles as seen in specific inverse trigonometric evaluations.