Problem 22 - Contemporary Precalculus

Question

Use a calculator in radian mode to approximate the functional value. $$\sin ^{-1}[\sin (-2)]$$

Step-by-Step Solution

Answer

Question: Approximate the functional value of \(\sin^{-1}[\sin (-2)]\) using a calculator in radian mode. Answer: ________ (in radians)

Step 1: Understand the problem

We need to find the functional value of \(\sin^{-1}[\sin (-2)]\). First, we need to recall that \(\sin^{-1}\) is the inverse sine function, and it's applied to the result of \(\sin(-2)\). Having a calculator in radian mode is necessary to obtain the correct result.

Step 2: Evaluate the sine function for the given value

Using the calculator in radian mode, evaluate the sine function for the given value, which is \(-2\). That is, find the value of \(\sin(-2)\).

Step 3: Apply the inverse sine function

Now that we have the value of \(\sin(-2)\), we will use the inverse sine function to find the answer. So, we want to find the value of \(\sin^{-1}[\sin (-2)]\) using the calculator.

Step 4: Interpret the result

Make sure to interpret the result obtained from the calculator in radian mode since the original problem asks to approximate the functional value in radians.

Key Concepts

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians relate the angle to the circumference of the circle. A circle has a total of \(2\pi\) radians. This makes radians especially useful in calculus and trigonometry, where they provide a more natural context for mathematical analysis.
When working with trigonometric functions, using radians can simplify computations and make it easier to understand the relationships between different math concepts.
For instance:

• Half a circle corresponds to \(\pi\) radians.
• The angle of 90° is \(\frac{\pi}{2}\) radians.

It is crucial to make sure your calculator is set to radian mode when solving problems that involve trigonometric functions like sine and cosine, especially when the problem specifies radians.

Sine Function

The sine function is one of the primary trigonometric functions and it relates a given angle to the ratio of the opposite side over the hypotenuse in a right triangle. However, in the unit circle, the sine function maps an angle directly to the y-coordinate of a point on the circle's circumference.
The range of the sine function is from -1 to 1. For any angle \(\theta\), the function outputs a value between these endpoints.
Key properties include:

• It is periodic with a period of \(2\pi\), meaning \( \sin(\theta) = \sin(\theta + 2\pi) \).
• The sine function is odd, so \( \sin(-\theta) = -\sin(\theta) \).

In the given exercise, you find \(\sin(-2)\), which requires understanding these properties to interpret the \(\sin^{-1}\) result correctly.
"Understanding these properties is essential, particularly when the inverse function is used to return results within the principal range, which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\)."

Calculator Usage

Using a calculator efficiently is key to successfully solving trigonometric problems, especially those involving inverse trigonometric functions in radians. Here's how you can make the most out of it:

• First, always ensure the calculator is in the correct mode. For the given problem, set it to radian mode as it affects how angles are interpreted.
• Recognize on your calculator the functions such as \("\sin^{-1}\", "\sin\", "\pi"\) and their placements on the calculator's keypad.
• For a function like \(\sin^{-1}[\sin(-2)]\), calculate \(\sin(-2)\) first, then use the inverse sine function to find the final answer.
• Re-check the results by considering the restrictions and range of the inverse function.

The calculator helps to find precise values quickly, but understanding the steps ensures correctness and meaning behind the numeric result.