Problem 77 - A Graphical Approach to Precalculus with Limits

Question

If \(n\) is an integer, \(n \cdot 180^{\circ}\) represents an integer multiple of \(180^{\circ},(2 n+1) \cdot 90^{\circ}\) represents an odd integer multiple of \(90^{\circ}\), and so on. Decide whether each expression is equal to \(0,1\), or \(-1\) or is undefined. $$\sin \left(n \cdot 180^{\circ}\right)$$

Step-by-Step Solution

Answer

The expression \( \sin(n \cdot 180^{\circ}) \) is equal to 0 for any integer \(n\).

Step 1: Understand the Sine Function Properties

The sine function, \( \sin(\theta) \), has specific values at well-known angles. The most pivotal angles related to multiples of \(180^{\circ}\) are \(0^{\circ}, 180^{\circ}, 360^{\circ}\), and so on. Here, \( \sin(0^{\circ}) = 0, \sin(180^{\circ}) = 0, \sin(360^{\circ}) = 0\), etc.

Step 2: Replace with Derived Angles

Given the expression \( \sin(n \cdot 180^{\circ}) \), we substitute the angle based on whether \(n = 0, 1, 2, \ldots\). Any integer multiple of \(180^{\circ}\) will be an exact angle like \(0^{\circ}, 180^{\circ}, 360^{\circ}\), etc.

Step 3: Evaluate the Sine at the Angles

Each resultant angle from \(n \cdot 180^{\circ}\) (such as \(0^{\circ}\), \(180^{\circ}\), \(360^{\circ}\), etc.) has the sine value of 0. Therefore, \( \sin(n \cdot 180^{\circ}) = 0 \) for any integer \(n\).

Step 4: Determine if the Expression is Defined

Since \( \sin(n \cdot 180^{\circ}) \) simplifies to \(0\) based on step 3 for all integer \(n\), the expression is defined and equals \(0\).

Key Concepts

Sine Function

The sine function, denoted as \( \sin(\theta) \), relates to the measure of an angle used within trigonometry. It is one of the basic trigonometric functions, alongside cosine and tangent.
The value of the sine function depends on the angle \( \theta \), usually measured in degrees or radians.
Typically, in a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse.

• For example, if an angle \( \theta = 30^{\circ} \), then \( \sin(30^{\circ}) = \frac{1}{2} \).
• For fundamental angle values such as \(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\), etc., sine values are either 0, 1, or -1.

The sine function exhibits periodic behavior, repeating its pattern every \(360^{\circ}\). This periodicity is crucial in understanding how sine functions behave for different angles, especially those involving integer multiples of \(180^{\circ}\).

Angle Measurement

Angles can be measured in degrees or radians, two systems that are often used interchangeably in mathematics.
In degrees, a full circle is \(360^{\circ}\). In contrast, in radians, a full circle is \(2\pi\) radians.
For trigonometry, degrees are often used for simplicity, especially in educational settings.

• A key conversion to remember is: \(180^{\circ} = \pi\) radians.
• This means \(n \cdot 180^{\circ}\) can also be written as \(n \cdot \pi\) radians.

Understanding angle measurement is essential because trigonometric functions like sine and cosine use these measures to determine their values.
Recognizing the equivalence of angles in degrees and radians can aid in calculations and in recognizing fundamental values.

Integer Multiples

Integer multiples refer to multiplying an integer value by a base number such as \(180^{\circ}\) in angles.
Here, \(n \cdot 180^{\circ}\) signifies taking an integer \(n\) and multiplying it by \(180^{\circ}\), resulting in angles like \(0^{\circ}\), \(180^{\circ}\), \(360^{\circ}\), and so forth.
Integer multiples are straightforward in concept but vital in understanding repeating cycles in trigonometric functions.

• For \(n = 0\), the angle is \(0^{\circ}\).
• For \(n = 1\), the angle is \(180^{\circ}\).
• For \(n = 2\), the angle is \(360^{\circ}\), and this continues as a pattern.

In trigonometry, these multiples are significant because they mark the points at which the sine function repeats its values.
For \(n \cdot 180^{\circ}\), the sine remains \(0\) for all integer \(n\). This results from the periodic nature of the sine function, making it predictable.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables.
These identities are crucial for simplifying expressions and solving trigonometric equations.
For angles that are integer multiples of \(180^{\circ}\), specific identities are particularly useful.

• \(\sin(0^{\circ}) = 0\)
• \(\sin(180^{\circ}) = 0\)
• \(\sin(360^{\circ}) = 0\) and this continues for any full cycle of \(180^{\circ} \cdot n\)

These identities help conclude that the sine of any multiple of \(180^{\circ}\) is zero.
By understanding and applying these trigonometric identities, you can simplify complex problems and unveil the symmetrical nature of the sine function on the unit circle.
This concept forms the basis for many practical applications in engineering, physics, and other scientific fields.