Problem 99 - A Graphical Approach to Precalculus with Limits

Question

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\sin \theta=\frac{1}{2}$$

Step-by-Step Solution

Answer

\( \theta = \arcsin\left(\frac{1}{2}\right) = 30^{\circ} \).

Step 1: Understanding the Problem

We need to find an acute angle \( \theta \) such that the sine of \( \theta \) is equal to \( \frac{1}{2} \). An acute angle is one that is less than \( 90^{\circ} \). Our goal is to express \( \theta \) as both an inverse trigonometric function and as a degree measure.

Step 2: Use Inverse Sine Function

To find the angle \( \theta \), we use the inverse sine function (also known as arcsin). The equation is: \[ \theta = \arcsin\left(\frac{1}{2}\right) \]. This function gives us the angle whose sine value is \( \frac{1}{2} \).

Step 3: Determine the Angle in Degrees

We know from trigonometry that the sine of \( 30^{\circ} \) is \( \frac{1}{2} \) because \( \sin(30^{\circ}) = \frac{1}{2} \). Thus, \( \arcsin\left(\frac{1}{2}\right) = 30^{\circ} \), which means \( \theta = 30^{\circ} \).

Key Concepts

Trigonometric Equations

Understanding trigonometric equations is a fundamental part of solving problems in trigonometry. Essentially, these equations involve trigonometric functions like sine, cosine, and tangent. When we solve these equations, we are typically searching for the angle or angles that satisfy the given equation.
A simple example could be an equation such as \( \sin \theta = \frac{1}{2} \). Our objective is to find the specific angle \( \theta \) that makes this equation true.

• Step 1: Identify which trigonometric function is present in the equation.
• Step 2: Use inverse trigonometric functions to solve for \( \theta \).
• Step 3: Determine if any additional conditions, like ensuring the angle is acute, are needed.

Acute Angles

An acute angle is any angle that measures less than \( 90^{\circ} \). In the context of trigonometry, we often deal with acute angles when determining sine, cosine, and tangent values.
Acute angles have properties that make them unique. For instance, their trigonometric ratios are all positive, which simplifies many equations and calculations.
For example, when we need to find \( \theta \) such that \( \sin \theta = \frac{1}{2} \), we are specifically looking for an acute angle. This condition points us specifically to \( \arcsin\left(\frac{1}{2}\right) \), or \( 30^{\circ} \).

• Always check that the solution fits within the range of acute angles.
• Remember that acute angles are often involved in standard triangle problems, making them useful for many geometric scenarios.

Degree Measure

The degree measure is a way to express angles using degrees, which is a common unit of measurement in mathematics. Degrees are part of a circular system of measurement where a full circle is \( 360^{\circ} \).
When solving the original problem \( \sin \theta=\frac{1}{2} \), we find that \( \theta \) is \( 30^{\circ} \). Here, the degree measure gives a clear, understandable value that represents the size of the angle.

• Most trigonometric functions and tables use degree measure as a standard.
• Conversion: Knowing how to convert between radians and degrees is useful, though in this specific problem, degrees are more straightforward.
• Degrees help in visualizing the angle more concretely in geometric settings.