Problem 11 - 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. \(\sin ^{-1} \frac{2}{3}\)

Step-by-Step Solution

Answer

\(\sin^{-1}\left(\frac{2}{3}\right)\approx 0.72973\) radians.

Step 1: Understand the Function

The problem involves finding the inverse sine of \(\frac{2}{3}\). The inverse sine function, denoted \( \sin^{-1} \), means we are looking for an angle whose sine is \(\frac{2}{3}\). The result will be in radians, since that's the default mode for most calculators.

Step 2: Use the Calculator

Turn on your calculator and make sure it is set to radian mode. Then, enter the function \(\sin^{-1}\left(\frac{2}{3}\right)\) into your calculator. This is usually done by entering \(2/3\), pressing the inverse sine function button (often represented as \( \sin^{-1} \) or \( \text{asin} \)).

Step 3: Approximate the Value

The calculator will display the value of \(\sin^{-1}\left(\frac{2}{3}\right)\) in radians. Record the value, ensuring it is approximate to five decimal places. For instance, you should get approximately 0.72973.

Key Concepts

Inverse Sine Function

The inverse sine function is a fundamental concept in trigonometry. Often denoted as \( \sin^{-1} \), this function is used to find the angle whose sine value is a known number. In simpler terms, if you know the sine of an angle, the inverse sine function helps you find what that angle is. For example, if you know that the sine of an angle is \( \frac{2}{3} \), using \( \sin^{-1} \) would tell you what angle in radians has that sine value.
The range of the inverse sine function is particularly important to remember. It outputs angles in the range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians. This ensures that for any given input within the range \(-1\) to \(1\), the function provides a unique output, remaining true to its definition as a function.
Understanding this concept allows you to translate real-world situations or geometric relationships into mathematical equations. For instance, knowing the height of a tree and the length of its shadow, you can determine the angle of elevation from the ground using inverse sine.

Radian Mode

Radian mode is a setting on calculators used commonly when dealing with trigonometric functions. It's important to understand that unlike degrees, which are based on dividing a circle into 360 parts, radians measure angles based on the radius of a circle. One complete revolution, a full circle, is \(2\pi\) radians.
By using radian mode, you ensure better compatibility with mathematical formulas and functions, especially in engineering, physics, and higher mathematics. Many calculators default to radian mode for this reason, ensuring consistency across trigonometric calculations.
When solving problems involving inverse sine, it's essential to ensure your calculator is set to radian mode. This will allow you to get a correct result that aligns with mathematical conventions. A simple check or toggle on your calculator settings will ensure that you are working in the right mode.

Approximation to Decimal Places

In mathematics, it's often necessary to approximate numerical values to a specific number of decimal places. This practice is crucial for maintaining precision while working with calculations, especially with irrational numbers.
For instance, when you calculate \(\sin^{-1}(\frac{2}{3})\), the value you obtain on your calculator may not exactly round off cleanly. Approximating it to five decimal places, such as 0.72973, ensures that you can use it efficiently in further calculations or report it with the desired precision.
When rounding, look at the sixth decimal place to decide if the fifth place rounds up or stays the same. For example, if the sixth place is 5 or more, increase the fifth decimal by one. This approach helps in making sure your calculations reflect a consistent level of precision, crucial in academic and professional settings.