Problem 78 - 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. $$\sin \left[(2 n+1) \cdot 90^{\circ}\right]$$

Step-by-Step Solution

Answer

The expression equals \(1\) or \(-1\) depending on if \((2n + 1) \cdot 90^\circ\) is a multiple of \(90^\circ\) leading to \(90^\circ\) or \(270^\circ\).

Step 1: Understanding the expression

We have the expression \( \sin[(2n + 1) \cdot 90^\circ] \). We need to evaluate this by considering the trigonometric properties of the sine function at specific angles.

Step 2: Recognize periodicity of sine

The sine function has a period of \(360^\circ\). This means \(\sin(\theta + 360^\circ k) = \sin(\theta)\) for any integer \(k\). We will find the value of the angle \((2n+1) \cdot 90^\circ\) within one full period.

Step 3: Reduce the angle modulo 360

\((2n + 1) \cdot 90^\circ\) simplifies to angles \(90^\circ, 270^\circ, 450^\circ, \ldots\). These angles are also congruent to \(90^\circ, 270^\circ\) modulo \(360^\circ\).

Step 4: Evaluate sine at specific angles

We need to evaluate \(\sin(90^\circ)\) and \(\sin(270^\circ)\). Using sine properties, \(\sin(90^\circ) = 1\) and \(\sin(270^\circ) = -1\).

Step 5: Draw conclusion for odd multiples

Since \(2n + 1\) is odd, \((2n + 1) \cdot 90^\circ\) results in either \(90^\circ\) or \(270^\circ\). For \(\sin(90^\circ)\), the result is \(1\), and for \(\sin(270^\circ)\), the result is \(-1\).

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that relates the angles of a right triangle to the ratios of its sides. It's commonly noted as \( \sin(\theta) \), where \( \theta \) is the angle in question. The sine function gives the y-coordinate of the point on the unit circle at a given angle. The function ranges from -1 to 1.

Depending on the angle, the sine can be zero, positive, or negative. Certain key angles produce specific sine values:

• \( \sin(0^\circ) = 0 \)
• \( \sin(90^\circ) = 1 \)
• \( \sin(180^\circ) = 0 \)
• \( \sin(270^\circ) = -1 \)

These output values are especially useful when dealing with various applications in science and engineering.

Periodicity

Understanding periodicity is key when working with trigonometric functions like sine. A function is periodic if it repeats values at regular intervals. For sine, this interval is \(360^\circ\). This means:

• \( \sin(\theta + 360^\circ) = \sin(\theta) \)
• \( \sin(\theta) \) has the same value every full rotation (every \(360^\circ\))

Due to periodicity, you can simplify expressions involving angles by reducing them modulo \(360^\circ\). For angles over \(360^\circ\), subtract \(360^\circ\) or its multiples until you find an equivalent angle within one complete rotation (0 to \(360^\circ\)). This approach helps in evaluating trigonometric expressions easily as demonstrated in the exercise.

Angles

Exploring the intricate world of angles can significantly enhance your understanding of trigonometric functions. Angles can be described in degrees or radians and represent the rotation about a point. In the context of the sine function, certain angles yield important results.

• Multiples of \(90^\circ\) are particularly noteworthy, as sine values are easily determinable at these points.
• An expression like \((2n+1) \cdot 90^\circ\) as used in the exercise, signifies an odd multiple of \(90^\circ\), leading often to calculations involving either \(90^\circ\) or \(270^\circ\).

When facing complex trigonometric tasks, reducing angles and making use of the periodic nature of the sine function simplifies calculations, just as shown by reducing congruent angles modulo \(360^\circ\), streamlining problem-solving efforts.