Problem 21 - Algebra and Trigonometry

Question

\(11-22\) . Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. \(\sin ^{-1}(-0.25713)\)

Step-by-Step Solution

Answer

The approximate value of \( ext{arcsin}(-0.25713)\) is \(-0.26002\) radians.

Step 1: Understanding the Problem

We need to find the inverse sine function, also known as arcsine, for the given value \(-0.25713\). This means we're looking for the angle whose sine is \(-0.25713\). The result should be in radians unless otherwise specified.

Step 2: Checking Validity of the Function

The function \( ext{arcsin}(x)\) is only defined for values \(-1 \leq x \leq 1\). Since \(-0.25713\) is within this range, the function is defined, and we can proceed to calculate.

Step 3: Using a Calculator

Input the value \(-0.25713\) into a calculator with the arcsin (or \( ext{sin}^{-1}\)) function to find the angle that corresponds to this sine value. Ensure the calculator is set to the correct mode (radians) to get the accurate result.

Step 4: Calculating the Result

Using the calculator, compute \( ext{arcsin}(-0.25713)\). The calculator should give an output, which is approximately \(-0.26002\) when rounded to five decimal places.

Key Concepts

Arcsine

The arcsine function is the inverse of the sine function. It helps us find the angle that produces a given sine value. With the symbol \( \sin^{-1} \) or sometimes \( \text{arcsin} \), it allows us to transition from a sine value back to its respective angle.
This function is only applicable within the range of the sine function,

• -1 to 1,

which means you can't find an angle for a sine value that's outside this range. Arcsine is especially useful when working with ratios from trigonometric calculations.
Always remember, the arcsine function will give you an angle in the range of

• \(-\frac{\pi}{2}\)
• to \(\frac{\pi}{2}\)

Angle Calculation

Calculating the angle using the arcsine function involves determining which angle corresponds to the sine value you have. Let's say we have

• the sine value of \(-0.25713\),

we need to find the angle whose sine gives this value.
The way this works is by using the inverse sine calculation, which can be done manually or, more practically, with a calculator. This is a direct method to navigate from a known sine value to its corresponding angle by following the reverse pathway of the sine function.
When you're working with arcsine, understanding the basic principles of inverse trigonometric functions is crucial for accurate calculations. Being familiar with the concept of inverse functions can significantly enhance your trigonometric skills.

Calculator Usage

Calculators play an essential role in modern trigonometric calculations, especially when dealing with inverse trigonometric functions like arcsine. When you want to find an angle using a calculator, make sure it has the capability of computing inverse functions.
Here’s a step-by-step guide to using your calculator for arcsine computations:

• First, confirm your calculator is set to the correct mode, typically radians for many trigonometric problems.
• Locate the arcsine function, usually denoted as \(\text{sin}^{-1}\) or \(\text{arcsin}\), in your calculator's function list.
• Input the given sine value, here \(-0.25713\).
• Press the calculate or equivalent button.
• Retrieve and round the result to the required decimal places, five in this example, resulting in approximately \(-0.26002\).

Using your calculator properly ensures efficient and accurate angle determination.

Radian Measure

In trigonometry, angles can be measured in degrees or radians. Radians offer a way to express angles based in terms of \(\pi\), which often simplifies calculations, especially when dealing with calculus or periodic functions.
When working with angles, understanding radian measure is crucial:

• One full rotation around a circle is \(2\pi\) radians.
• A half rotation is \(\pi\) radians, which covers 180 degrees.
• The arcsine function gives you an angle measured in radians, so when entering a value or expecting a result, watch that your calculator is in the correct mode.

Using radian measure aligns calculations more naturally with the mathematics of circular motion and periodic functions, making complex trigonometric problems much simpler to handle.