Problem 20 - Precalculus Essentials

Question

Use a calculator to find the value of each expression rounded to two decimal places. $$\sin ^{-1} 0.47$$

Step-by-Step Solution

Answer

Since this calculation result differs between calculators, since some return degrees and others return radians, the specific answer cannot be given. After the button 'equals' is pressed, the displayed number should be the final answer rounded to two decimal places.

Step 1: Set up the function

Input the expression, \(\sin^{-1}(0.47)\), into the calculator. Here, \(\sin^{-1}\) is the inverse sine or arcsine function and 0.47 is the value whose arcsine you need to find.

Step 2: Perform the Operation and Round

After you've inputted the expression into the calculator, press the 'equals' button. The calculator will perform the operation and display a result. Round the displayed result to the nearest two decimal places. For example, if the result is 0.123456789, you should round it to 0.12.

Key Concepts

Arcsine Function

Understanding the arcsine function, commonly noted as \( \sin^{-1} \) or arcsin, is crucial when dealing with trigonometry. In simple terms, the arcsine function is the inverse of the sine function. While the sine function gives you the ratio of the opposite side to the hypotenuse in a right triangle for a given angle, arcsine function does the reverse; it provides you with the angle when you already know that ratio.

For instance, if we have the ratio 0.47, the arcsine function, expressed as \( \sin^{-1}(0.47) \) or arcsin(0.47), gives us the angle whose sine is 0.47. This operation is especially important in fields such as physics and engineering where angles often need to be determined from various ratios.

To find the arcsine using a calculator, you usually have to switch it to scientific mode since basic calculators typically won't provide this function. It's important to note that the result will be in radians or degrees depending on the settings of your calculator, so double-check this before proceeding.

Calculating Trigonometric Functions

Trigonometric functions are the cornerstone of understanding relationships in triangles, specifically right-angled triangles, and they have applications in waves, oscillations, and many areas of science and engineering. Apart from the sine function, there are other trigonometric functions like cosine, tangent, and their respective inverse functions, including arcsine. Calculating these functions often requires a scientific calculator or trigonometric tables.

When calculating the value of trigonometric functions, precision is key, especially in the sciences, where a small error in calculation can result in significant discrepancies in real-world applications. Hence, being familiar with the functions and the process to derive them is essential for accuracy in both academic and professional environments.

Rounding Decimals

Rounding decimals is an important mathematical skill that helps simplify numbers to make them easier to work with, especially for estimations or when a certain degree of precision is required. When you round to two decimal places, you look at the third decimal place to decide whether to round up or down. If the third decimal place is 5 or above, you round up; if it's 4 or below, you round down.

This is particularly important when you're working with measurements, financial calculations, or when the exact value is not necessary. For example, if you have a calculator result of 0.123456789 and need to round it to two decimal places, you consider the third decimal place, which is 3. Since it is below 5, the number rounds down to 0.12. Correct rounding is essential to ensure that your final answer is both accurate and appropriate for the situation.