Problem 16 - 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. \(\sin ^{-1}(0.13844)\)

Step-by-Step Solution

Answer

The approximate value of \( \sin^{-1}(0.13844) \) is 0.13868 radians.

Step 1: Understand the Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1} \), is used to find an angle whose sine value is a given number. It is important to remember that the result of \( \sin^{-1}(x) \) is an angle, usually expressed in radians, and it lies within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 2: Use a Calculator to Evaluate

Enter the value 0.13844 into your calculator, using the inverse sine function (often labeled as "asin" or "sin^(-1)" on calculators). Ensure the calculator is set to radian mode to get the angle in radians.

Step 3: Record the Result

The calculator should display a result close to 0.13868 radians. This value represents the angle whose sine is 0.13844, rounded to five decimal places as required by the exercise.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in relation to angles and triangles. They describe the relationships between the angles and sides of right triangles. The main trigonometric functions you will encounter are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These are essential in not only geometry, but also in physics, engineering, and other sciences.Inverse trigonometric functions are a critical extension of these concepts. They allow us to determine the angle when given a trigonometric ratio. For instance, the inverse sine function (\(\sin^{-1}\)) helps us find an angle \(\theta\) for which \(\sin(\theta) = x\). This inverse function is key for solving equations where the angle is the unknown. The range of \(\sin^{-1}\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), meaning it gives an angle within this interval. This range ensures that for any given value, there is a single, consistent angle output, maintaining the function's properties.

Radian Mode

In trigonometry, angles can be measured in degrees or radians. Radians are often used in higher mathematics because they are a natural unit of measure based on the arc length of a circle. One full circle is \(2\pi\) radians, which is equivalent to 360 degrees. To better understand this, consider that \(\pi\) radians equals 180 degrees.When dealing with trigonometric functions, especially in a calculus or physics setting, you should become comfortable with radians. Most scientific calculators have a setting for radian mode, as well as degree mode. Radian mode allows your calculator to interpret angles correctly when evaluating trigonometric functions.Ensuring the calculator is set to radian mode is crucial when solving problems involving inverse trigonometric functions. This, as with the exercise example, ensures that when using \(\sin^{-1}(x)\), the resulting angle will rightly be in radians, providing accurate and expected results.

Angle Calculation

Calculating angles using inverse trigonometric functions is an essential skill. This involves using tools like calculators to efficiently find precise angle values, given trigonometric ratios.To calculate an angle using \(\sin^{-1}(x)\), follow these steps:

• First, determine if the value is within the valid range for the sine function, which is \([-1, 1]\).
• Next, using a calculator, enter the value and select the inverse sine function, often labeled "asin" or "sin^(-1)". Make sure your calculator is in radian mode if the problem specifies radians.
• The calculator will output the angle in radians, which you then round to the desired precision. For example, solving \(\sin^{-1}(0.13844)\) yields approximately 0.13868 radians, to five decimal places.

This precise calculation helps in geometry, physics, and engineering problems where exact angle measurements are crucial. Understanding these steps ensures that you'll be able to tackle any problem involving trigonometric angle calculations.