Problem 35 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\sin \left(2 \sin ^{-1} \frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The value is approximately 0.87.

Step 1: Understanding the Problem

We need to evaluate the expression \(\sin \left(2 \sin^{-1} \frac{1}{2}\right)\). This involves finding the inverse sine of \(\frac{1}{2}\), doubling the result, and then taking the sine of that value.

Step 2: Evaluate the Inverse Sine Part

First, we need to find \(\sin^{-1}\left(\frac{1}{2}\right)\). The inverse sine of \(\frac{1}{2}\) gives us the angle whose sine is \(\frac{1}{2}\), which is \(\frac{\pi}{6}\) radians.

Step 3: Double the Angle

Next, we take the result from the inverse sine, \(\frac{\pi}{6}\), and double it to find \(2 \times \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}\).

Step 4: Apply the Sine Function

Finally, we compute \(\sin\left(\frac{\pi}{3}\right)\). The sine of \(\frac{\pi}{3}\) is \(\frac{\sqrt{3}}{2}\). To round to the nearest hundredth, \(\frac{\sqrt{3}}{2} \approx 0.87\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics, especially when we need to determine angles from given trigonometric values. When dealing with the inverse sine function, denoted as \(\sin^{-1}(x)\), it allows us to find an angle \(\theta\) such that the sine of that angle is \(x\). In our exercise, we determined \(\sin^{-1}\left(\frac{1}{2}\right)\). This means we sought the angle whose sine is \(\frac{1}{2}\). This specific angle is \(\frac{\pi}{6}\) radians, or 30 degrees, in the first quadrant where the sine function is positive.

Inverse trigonometric functions are defined only within specific domains to ensure they return unique results. For sine, this is typically from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians, allowing us to solve equations like \(\sin\theta = \frac{1}{2}\) confidently. It's crucial to understand these domains while evaluating or applying inverse functions, as they ensure our solutions remain "principal values" - angles found within these ranges.

This understanding is directly applicable to countless practical fields, such as engineering and physics, where determining precise angles from sine, cosine, or tangent values is commonplace.

Angle Measurement

Angles are a fundamental measure of rotation and are typically measured in either degrees or radians. It's essential to grasp these units and how they relate to one another, especially when dealing with trigonometric functions. The radian, a key measurement in our problem, measures angles based on the radius of a circle.

One full rotation around a circle is \(2\pi\) radians, which equals 360 degrees. Therefore, \(\frac{\pi}{6}\) radians, which was used in our inverse sine calculation, equates to 30 degrees. Measuring angles with radians is often preferred in mathematics and physics because it simplifies many formulas and calculus derivations.

Understanding and using radians ensures accuracy, particularly when calculating arc lengths and areas in geometry, making this unit indispensable in advanced studies. Practical applications like navigation and computer graphics often rely on radian measures to calculate smooth and accurate rotations.

Radian Conversion

Converting between degrees and radians is straightforward, and understanding this conversion is vital for solving problems like the one in our exercise. The conversion factor between degrees and radians is crucial: \(180\) degrees is equivalent to \(\pi\) radians.

To convert degrees to radians, you multiply the degree measure by \(\frac{\pi}{180}\). Conversely, converting radians to degrees involves multiplying the radian measure by \(\frac{180}{\pi}\). For example, to convert \(\pi/3\) radians (as obtained when doubling our angle \(\frac{\pi}{6}\)) to degrees, we compute \(\frac{\pi}{3} \times \frac{180}{\pi} = 60\) degrees.

This conversion is not only a matter of academic exercise but also an important skill whenever you switch between systems in engineering, computer science, or physics tasks. By navigating fluently between radians and degrees, you ensure precision while accommodating different standards and contexts across disciplines.