Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. arcsin 0.45

Step-by-Step Solution

Answer

The approximate value of \(\sin^{-1}(0.45)\) to the nearest hundredth is 26.57 degrees.

Step 1: Understand the Notation

Arcsin or \(\sin^{-1}\) refers to the inverse of the sine function. We are asked to find the value of \(\sin^{-1}(0.45)\).

Step 2: Use Calculator

Use a scientific calculator to find the value of \(\sin^{-1}(0.45)\). It's important to make sure the calculator is in the correct mode (degrees or radians) as per the requirement. Most calculators are set to degrees, which is the mode we can use in absence of specific instructions.

Step 3: Round the Answer

The result obtained needs to be rounded to the nearest hundredth as per the question's instruction. If the value on the third place after decimal is 5 or higher, add 1 to the value at the second decimal place.

Key Concepts

Scientific Calculator

A scientific calculator is a handy tool for solving various mathematical problems, especially those involving complex operations like trigonometric functions. These calculators are equipped with functions such as sine, cosine, tangent, and their inverses, such as arcsin, arccos, and arctan. When dealing with inverse trigonometric functions, like arcsine, a scientific calculator allows you to input the value directly, such as \(\sin^{-1}(0.45)\), and it returns the angle whose sine is 0.45.

To use a scientific calculator effectively:

• Make sure your calculator is in the correct mode (degrees or radians). This choice affects the output value significantly. Most often, calculations are done in degrees.
• Locate and press the sin extsuperscript{-1} or arcsin button.
• Enter the given value, like 0.45, and execute the calculation.

By following these steps, you quickly find the inverse and can proceed with further calculations, such as rounding.

Rounding Numbers

Rounding numbers makes them simpler by reducing the number of decimal places, while keeping the value close to what it was. In many situations, like exams or in daily life, you might be asked to round your result to a specific decimal place. In this exercise, the requirement is to round to the nearest hundredth, which means two decimal places, such as turning 1.2345 to 1.23.

For rounding to the nearest hundredth:

• Look at the third decimal place. If it's 5 or more, increase the second decimal place by 1.
• If the third decimal place is less than 5, leave the second decimal as it is.
• For example, 0.231 becomes 0.23, but 0.235 becomes 0.24.

By applying these simple rules, you can ensure that your answer is both accurate and presented in the desired format.

Arcsine Function

The arcsine function, often written as arcsin or \(\sin^{-1}\), is the inverse operation of the sine function. This function is used to determine the angle whose sine value is a given number. For example, if you have \(\sin^{-1}(0.45)\), you're looking for an angle that has a sine of 0.45.

The property of arcsin is crucial because:

• It allows you to switch between a sine value and its corresponding angle.
• Understanding arcsin helps in solving trigonometric equations, especially in cases where angles need to be calculated directly.
• The principal value range of arcsin is between -90° and 90° (or -\(\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians), providing a specific interval within which the solution lies.

Knowing how to find arcsin using your calculator empowers you to tackle a variety of math problems efficiently, ensuring a solid grasp of trigonometric functions.