Problem 24 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{25 \pi}{2}\) (b) \(\cos \frac{25 \pi}{2}\) (c) \(\cot \frac{25 \pi}{2}\)

Step-by-Step Solution

Answer

(a) 1, (b) 0, (c) 0

Step 1: Understand the periodicity of sine and cosine

The sine and cosine functions have a period of \(2\pi\). This means that \(\sin(\theta + 2\pi k) = \sin(\theta)\) and \(\cos(\theta + 2\pi k) = \cos(\theta)\) for any integer \(k\).

Step 2: Reduce the angle for sine

Since \(\sin\) is periodic with period \(2\pi\), we reduce \(\frac{25\pi}{2}\) by determining the equivalent angle within one full period. Divide \(\frac{25\pi}{2}\) by \(2\pi\) to find the integer quotient: \(\frac{25\pi}{2} \div 2\pi = 6.25\). This means 6 full periods, leaving us with a remainder of \(\frac{\pi}{2}\).

Step 3: Calculate \(\sin \frac{25\pi}{2}\)

Given \(\frac{25\pi}{2}\) is equivalent to \(\frac{\pi}{2}\) within one period of \(\sin\), we have: \(\sin \frac{25\pi}{2} = \sin \frac{\pi}{2} = 1\).

Step 4: Reduce the angle for cosine

Similarly, we reduce \(\frac{25\pi}{2}\) by its periodicity with \(2\pi\). As calculated previously, this leaves us with the angle \(\frac{\pi}{2}\).

Step 5: Calculate \(\cos \frac{25\pi}{2}\)

With \(\frac{25\pi}{2} \equiv \frac{\pi}{2}\), we evaluate \(\cos \frac{\pi}{2} = 0\).

Step 6: Calculate \(\cot \frac{25\pi}{2}\)

The cotangent is defined as \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Using \(\cos \frac{25\pi}{2} = 0\) and \(\sin \frac{25\pi}{2} = 1\), we find \(\cot \frac{25\pi}{2} = \frac{0}{1} = 0\).

Key Concepts

Periodicity and Trigonometric Functions

Periodicity in trigonometric functions, specifically sine and cosine, is a crucial concept. Both functions repeat their values in regular intervals, known as their period. For sine and cosine, this period is every \(2\pi\) radians. This property helps in reducing any angle to an equivalent angle within the range of \(0\) to \([2\pi)\) or to simplify calculations within one full circle.

• For instance, if you have an angle like \(\frac{25\pi}{2}\), you can find an equivalent angle by dividing by \(2\pi\).
• The integer part of this division indicates how many full periods (or circles) the angle has completed.
• The remainder tells us the equivalent angle within the standard circle.

This reduction not only simplifies evaluation but aligns angles within one complete cycle, making calculations manageable.

Sine and Cosine Properties

Understanding sine and cosine properties involves their behavior over their periodic cycle. For sine, the values range from \(-1\) to \(1\).

• At \(\frac{\pi}{2}\), \( ext{sin}\) reaches its maximum, which is \(1\).
• It implies that when working with \(\frac{25\pi}{2}\), we simplify this to \(\frac{\pi}{2}\), based on periodicity, yielding \(\text{sin}\,\frac{25\pi}{2} = 1\).

The cosine function, sharing the same period, has its maximum value at \(0\) and again starts from \(0\) at \(\frac{\pi}{2}\):

• This behavior results in \(\text{cos}\) equaling \(0\) at both \(\frac{\pi}{2}\) and, after reducing, \(\frac{25\pi}{2}\).
• Such periodic properties are essential for solving trigonometric values of large angles without tedious recalculations.

Cotangent Definition and Specific Evaluation

The cotangent function defines itself as the ratio of cosine to sine, or \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). This definition is essential to understand when the sine is \(1\) at angles like \(\frac{25\pi}{2}\).

• Since \(\cos \frac{\pi}{2} = 0\) and \(\sin \frac{\pi}{2} = 1\), we find \(\cot \frac{25\pi}{2} = 0\).
• The evaluation demonstrates that when the cosine is zero, leading to \(\cot \theta\) being zero, it signifies the angle is tangential to the sine peak.

This property provides insight into how trigonometric identities interact and offers shortcuts in calculating trigonometric values. Learning cotangent's relation to sine and cosine gives a more comprehensive view of trigonometric functions.