Problem 4 - Algebra 2 and Trigonometry

Question

For each function value, write the value or tell why it is undefined. Do not use a calculator. \(\cos \frac{\pi}{2}\)

Step-by-Step Solution

Answer

The value is 0 because the x-coordinate at \(\frac{\pi}{2}\) radians is 0.

Step 1: Understanding the cosine function

The cosine function represents the adjacent side over the hypotenuse in a right triangle or can be viewed on the unit circle as the x-coordinate of a point that an angle intercepts on the circle. The cosine function is periodic with period of \(2\pi\).

Step 2: Locate the angle on the unit circle

The angle \(\frac{\pi}{2}\) radians corresponds to 90 degrees on the unit circle. At this angle, the point on the unit circle is located at the top of the circle, which has coordinates \((0, 1)\).

Step 3: Determine the cosine value

Since cosine of an angle in the unit circle is the x-coordinate of the corresponding point, for the angle \(\frac{\pi}{2}\), the x-coordinate is 0.

Key Concepts

Unit Circle

The unit circle is a powerful and fundamental tool in trigonometry. It is named the "unit circle" because it is a circle with a radius of exactly one unit. Positioned on a coordinate plane, its center is located at the origin (0, 0). This allows for easy understanding of trigonometric functions. In terms of the unit circle:

• Any point on the circle can be described using coordinates \( (x, y) \).
• The angle formed by a line drawn from the origin to this point is measured in radians.
• This circle helps calculate the sine and cosine of angles by representing them as the y and x coordinates, respectively, of points on the circle.

When we consider an angle such as \( \frac{\pi}{2} \), we can find it by visualizing this angle as a rotation from the positive x-axis. This position is directly at the top of the circle, represented by the coordinates \( (0, 1) \). Understanding this visual aspect makes solving problems involving trigonometric functions much more intuitive.

Trigonometric Functions

Trigonometric functions are essential mathematical concepts that relate to angles in various mathematical contexts, such as geometry, physics, and engineering. In particular, the cosine function, \( \cos(\theta) \), relates an angle's measure to the x-coordinate of a point on the unit circle. Similarly, the sine function, \( \sin(\theta) \), relates it to the y-coordinate. These functions are periodic, meaning they repeat their values in regular intervals. For cosine, this period is \( 2\pi \), which signifies that the function's values start repeating after this interval. Key Characteristics of the Cosine Function:

• It is even, meaning \( \cos(-\theta) = \cos(\theta) \).
• The function's range is between -1 and 1, inclusive.
• It equals 0 at angles such as \( \frac{\pi}{2} \) because the x-coordinate at these angles is 0.

Grasping these properties allows students to efficiently compute angles and understand their geometric implications without the assistance of calculators.

Angle Measurement

Angles can be measured in different units, predominantly degrees and radians, each offering unique advantages for various applications. Understanding these measurements is crucial for effectively working with trigonometric functions.Radian measure is central to calculus and analysis, as it provides a natural way of describing periodic phenomena such as cycles and rotations in the unit circle. Key Insights into Angle Measurements:

• One full circle, or a complete rotation in radians, is \( 2\pi \).
• In degrees, this same full circle is 360 degrees.
• Therefore, an angle of \( \frac{\pi}{2} \) radians, which corresponds to 90 degrees, represents a quarter turn.

Switching between these units involves straightforward conversions: \( \pi \) radians equals 180 degrees. Thus, understanding both measurements enables one to appreciate their distinct roles in trigonometry and broader mathematics, aiding in the visualization and calculation within the unit circle context.