Problem 15 - Calculus: Early Transcendentals

Question

Consider \(f(x)=\sin x .\) What happens when we choose \(x_{0}=\pi / 2 ?\) Explain.

Step-by-Step Solution

Answer

The function evaluates to 1.

Step 1: Setting up the function

Given the function is \(f(x) = \sin x\). We need to find the value of \(f(x_0)\), where \(x_0 = \frac{\pi}{2}\).

Step 2: Substituting into the function

Substitute \(x = \frac{\pi}{2}\) into the function. This gives us \(f \left( \frac{\pi}{2} \right) = \sin \frac{\pi}{2}\).

Step 3: Evaluating the sine function at \(x = \frac{\pi}{2}\)

The sine of \(\frac{\pi}{2}\) is 1. Hence, \(f \left( \frac{\pi}{2} \right) = 1\).

Step 4: Conclusion

Thus, when choosing \(x_0 = \frac{\pi}{2}\), the function \(f(x) = \sin x\) evaluates to 1.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that relates to the position of angles in the unit circle. It is commonly denoted as \( \sin x \). The sine of an angle \( x \) gives the vertical coordinate of the corresponding point on the unit circle.
This function is periodic with a period of \( 2\pi \) radians, or 360 degrees. This means it repeats its values in regular intervals. The sine function oscillates smoothly between -1 and 1.
For example:

• \( \sin(0) = 0 \)
• \( \sin(\pi/2) = 1 \)
• \( \sin(\pi) = 0 \)
• \( \sin(3\pi/2) = -1 \)
• \( \sin(2\pi) = 0 \)

The sine function is vital in many fields, including physics and engineering, because it models oscillatory behavior like waves and sound.

Evaluating Trigonometric Values

Evaluating trigonometric values means determining the specific value of a trigonometric function for a given angle. This process often involves knowing certain trigonometric properties and identities.
For example, when the task is to evaluate \( \sin \frac{\pi}{2} \), you follow these steps:

• Identify the angle in either degrees or radians. Here, \( \frac{\pi}{2} \) is used.
• Recall key trigonometric values or use the unit circle.
• For \( \sin \frac{\pi}{2} \), it's a known value that equals 1, indicating the point (0,1) on the unit circle.

To make this process simpler, familiarize yourself with common angles like 0, \( \pi/2 \), \( \pi \), \( 3\pi/2 \), and \( 2\pi \). These reference angles often make calculating exact values straightforward. Using the unit circle as a visual tool can help in understanding these values intuitively.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. These identities help simplify complex trigonometric expressions and are essential tools in algebra and calculus.
Common identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Angle Sum and Difference identities:
• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)
• Double Angle formula: \( \sin(2x) = 2\sin x \cos x \)

Using these identities can transform and simplify expressions, allowing for easier problem-solving in trigonometry. They are particularly useful in proving other mathematical properties and are widely used in mathematical analyses involving periodic functions.