Question

Use a calculator to approximate each value. \(\sin ^{-1}(0.1113)\)

Short answer

\( \sin^{-1}(0.1113) \approx 6.397^{\circ} \) or \( 0.1115 \) radians.

Step-by-step solution

Understand the Problem

We need to find the angle whose sine is 0.1113. This is represented as \( \sin^{-1}(0.1113) \), which is an inverse trigonometric function (also called arcsine).

Use a Calculator

Enter 0.1113 into your calculator and find the inverse sine function. Make sure your calculator is set to degrees if you want the answer in degrees, or radians if you prefer that measure.

Read the Calculator Output

After calculating \( \sin^{-1}(0.1113) \), the calculator should display the result. Depending on the mode (degrees or radians), note down the value shown.

Interpret the Result

The value displayed on the calculator is the angle that has a sine of 0.1113. This is the approximate angle we were asked to find.

Key concepts

Understanding Arcsine

When we talk about the arcsine function, it helps us answer: "What angle gives me this sine value?" The notation \( \sin^{-1}(x) \) or "arcsine of \( x \)" denotes an inverse trigonometric function that finds the angle whose sine is \( x \). In simple terms, if \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \).
It's crucial to note that the function arcsine returns values within specific ranges. For example, in most calculators, the arcsine function returns values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) if you're using radians, or between -90° and 90° for degrees. This range ensures that the function is properly defined. Any angle outside this range might not be directly given by the calculator unless you manually adjust it using trigonometric identities.
This concept is particularly useful in scenarios where we know a ratio from a right triangle – such as the opposite side over the hypotenuse – and we want to find the angle itself.

• Sine is the ratio of the opposite side to the hypotenuse in a right triangle.
• Arcsine helps us find that angle corresponding to a given sine value.

Using a Calculator for Arcsine

Utilizing a calculator for finding arcsine values can save time and effort, but knowing the right steps is crucial. First, identify the value whose arcsine you need, like \( 0.1113 \) in this exercise. Afterward, ensure your calculator is set to the correct mode. For angles, it's typically required to set the calculator to either degrees or radians, depending on your preference or what's asked in the problem.

• Turn on your calculator: Ensure it's charged or has fresh batteries.
• Check the mode: Decide whether you want the angle in degrees or radians. Most calculators have a button to toggle between these modes.
• Enter the value: Type in the value (e.g., \( 0.1113 \)).
• Access the inverse function: This might be a secondary function on the same key as sine, often accessed by pressing a "shift" or "2nd" button first.
• Read the result: Once calculated, note whether the presented angle is in the selected units (degrees or radians).

By following these steps, the calculator should quickly give you the desired approximation of the angle. Always double-check your settings and inputs to ensure accuracy.

Understanding Angle Approximation

In trigonometry, angle approximation often involves determining the nearest value of an angle from a given trigonometric function value, like the arcsine in this problem. Approximation is essential, especially when the exact angle might not have a simple, small decimal representation. Here's what to keep in mind when approximating angles:

• Significance of decimal points: With sine values, small changes in the angle might lead to different decimal results; precision in your calculator's settings can ensure accurate approximation.
• Rounding: Calculators might round angles to a certain decimal place. Be mindful if the problem requires a higher degree of precision.
• Inherent limits: Calculators have limits based on their display capabilities, which means they cannot present more decimals than they are programmed to.

When working with approximations, remember that they provide near values useful for practical scenarios, such as construction or navigation, where exact measurement might be less critical. Always ensure your approximation aligns with the context and precision requirements of your problem.