Problem 9 - Algebra 2

Question

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

Step-by-Step Solution

Answer

0.58

Step 1: Find the angle in radians using arcsin

We start by finding the angle, denoted as \( \theta \), such that \( \sin(\theta) = \frac{1}{2} \). This angle is \( \theta = \sin^{-1}\left(\frac{1}{2}\right) \). In radians, \( \theta = \frac{\pi}{6} \) because \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \).

Step 2: Calculate the tangent of the angle

Now, we need to find \( \tan(\theta) \) where \( \theta = \frac{\pi}{6} \). We know that \( \tan\left(\frac{\pi}{6}\right) = \frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)} \). Since \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \) and \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \), this simplifies to: \[ \tan\left(\frac{\pi}{6}\right) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \].

Step 3: Simplify and round the result

We simplify the expression \( \frac{1}{\sqrt{3}} \) by rationalizing it. This gives us \( \frac{\sqrt{3}}{3} \). Finally, we round \( \frac{\sqrt{3}}{3} \) to the nearest hundredth. The value of \( \sqrt{3} \approx 1.732 \), so \( \frac{\sqrt{3}}{3} \approx 0.577 \), which is rounded to 0.58.

Key Concepts

Radians Conversion

Converting degrees to radians is a fundamental skill in trigonometry. Degrees measure how much an angle opens, while radians measure this angle relative to the radius of a circle. One complete rotation around a circle equals 360 degrees, which is equivalent to \(2\pi\) radians. Therefore, \(1\) radian is \(\frac{180}{\pi}\) degrees.To convert from degrees to radians, we use the formula:

• \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\)

In the exercise, \(\sin^{-1}(\frac{1}{2})\) equals \(\frac{\pi}{6}\) radians, which corresponds to 30 degrees because the sine of 30 degrees is \(\frac{1}{2}\). Remember, working with radians is often preferred in calculus and advanced mathematics due to their natural relation to the unit circle.

Tangent Calculation

Calculating the tangent of an angle involves understanding how the tangent function relates to other trigonometric functions. The tangent of an angle \(\theta\) is defined as the ratio of the sine of the angle to the cosine of the angle, expressed as:

• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

In this problem, we found that \(\theta = \frac{\pi}{6}\). Thus, \(\tan\left(\frac{\pi}{6}\right)\) is calculated as \(\frac{1/2}{\sqrt{3}/2}\), simplifying to \(\frac{1}{\sqrt{3}}\).This process involves using the values:

• \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)
• \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)

Understanding these relationships is crucial, especially in problems that interconnect multiple trigonometric functions.

Rationalizing Denominators

Rationalizing denominators is an algebraic technique used to eliminate the square roots from the bottom part of a fraction. This is done to simplify expressions and make them easier to understand and use in further calculations.Consider the fraction \(\frac{1}{\sqrt{3}}\). To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{3}\):

• \(\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)

The denominator \(\sqrt{3} \times \sqrt{3}\) becomes 3, a rational number, which is the goal of rationalization.This step not only simplifies the expression but also helps in approximating values more easily. For instance, \(\frac{\sqrt{3}}{3}\) approximates to 0.577, and when rounded to the nearest hundredth, it becomes 0.58, making it more straightforward in applications.