Problem 19 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\sin 30^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \( \sin 30^{\circ} \) is \( \frac{1}{2} \), which is rational.

Step 1: Identify the Expression

We are given the expression \( \sin 30^{\circ} \). The task is to find the exact value of this trigonometric function.

Step 2: Use Known Values of Sine

From trigonometric values, it's known that \( \sin 30^{\circ} = \frac{1}{2} \). This is a well-known result that's derived from the unit circle.

Step 3: Check for Irrationality

Since \( \frac{1}{2} \) is a rational number (it can be expressed as a fraction of integers), there is no need to perform a decimal approximation to support its irrationality. The exact value is rational.

Key Concepts

Sine Function

The sine function is a fundamental part of trigonometry used to find the ratio of the opposite side to the hypotenuse in a right triangle. It's denoted as \( \sin \theta \), where \( \theta \) is an angle. Understanding the sine function is essential for converting between angles and sides of a triangle.

The sine function is particularly significant in the unit circle, which is a different approach to understanding trigonometric functions globally, beyond just the triangle. In the context of the unit circle, the sine of an angle is defined as the \( y \)-coordinate of the point on the unit circle corresponding to that angle.

• The sine function is periodic with a period of \( 360^{\circ} \) or \( 2\pi \) radians.
• It reaches its maximum value of 1 and minimum value of -1.
• The unit circle allows us to comprehend the sine of any angle, even those beyond \( 0^{\circ} \) and \( 360^{\circ} \).

Unit Circle

The unit circle is a crucial concept in trigonometry that helps us understand how trigonometric functions behave in a two-dimensional plane. It is a circle with a radius of 1 centered at the origin of the coordinate system. The unit circle offers a visual representation of the functions, including sine and cosine, as circular projections.

To link the unit circle to the sine function, consider any given angle. Place this angle in standard position with its vertex at the circle's origin. The point where the terminal side of the angle intersects the unit circle provides the sine and cosine values.

• For \( \sin 30^{\circ} \), the point on the unit circle is \((\frac{\sqrt{3}}{2}, \frac{1}{2})\).
• The \( y \)-value of this point, \( \frac{1}{2} \), gives us the sine of \( 30^{\circ} \).
• The unit circle makes it intuitive to see why \( \sin 30^{\circ} \) equals \( \frac{1}{2} \).

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, with the denominator \( q \) not equal to zero. They form an important subset of real numbers because they include both whole numbers and decimals that terminate or repeat.

When evaluating trigonometric functions like \( \sin 30^{\circ} \), knowing whether the result is rational or irrational is crucial. In this case, \( \sin 30^{\circ} = \frac{1}{2} \), a rational number clearly expressed as a simple fraction.

• Rational numbers include familiar fractions like \( \frac{1}{2} \), \( \frac{3}{4} \), and whole numbers like 1, 2, 3, etc.
• They differ from irrational numbers, which cannot be expressed as a simple fraction. Examples include \( \pi \) and \( \sqrt{2} \).
• Knowing the nature of a number helps determine whether further steps, like decimal approximation, are required.