Problem 84 - Precalculus

Question

For the following exercises, find the exact value of each trigonometric function. $$ \sin \frac{\pi}{2} $$

Step-by-Step Solution

Answer

\( \sin \frac{\pi}{2} = 1 \)

Step 1: Recognize the Angle

The angle given is \( \frac{\pi}{2} \), which is a special angle on the unit circle.

Step 2: Locate the Angle on the Unit Circle

The angle \( \frac{\pi}{2} \) corresponds to 90 degrees on the unit circle. It is located at the top of the circle where the y-coordinate of the point is 1 and the x-coordinate is 0.

Step 3: Identify the Sine Function Value

In the unit circle, the sine of an angle is equal to the y-coordinate of the corresponding point. Since the y-coordinate of the point at \( \frac{\pi}{2} \) is 1, the value of \( \sin \frac{\pi}{2} \) is 1.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of exactly one unit, centered at the origin of a coordinate plane. A unique feature of the unit circle is its ability to simplify the understanding and calculation of trigonometric functions.In the coordinate plane, the unit circle is represented by the equation \( x^2 + y^2 = 1 \).Why is it called the "unit" circle? Because its radius is one! This simplicity makes it easier to relate angles to coordinates:

• The circumference is divided into radians, which are a way of expressing angles.
• Every point on the circle corresponds to specific \( (x, y) \) coordinates.

Each angle on the unit circle corresponds to a point, which assists in visualizing and calculating trigonometric functions. For example, when an angle, measured in radians, is plotted on the unit circle, its corresponding point \( (x, y) \) is utilized to determine the sine and cosine values directly.

Special Angles

Special angles are specific angles often used in trigonometry that facilitate easy calculation and understanding of trigonometric functions. These angles typically include \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \(\frac{\pi}{2}\), corresponding to \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\).Here's why special angles are important:

• They have well-known exact values for sine, cosine, and tangent functions, which you can memorize or utilize quickly during problem-solving.
• They often appear in math problems due to their ease of calculation and frequent occurrence in real-world phenomena.

For example, the angle \(\frac{\pi}{2}\) is equivalent to 90 degrees, placing it at the topmost point of the unit circle where the coordinates are \((0, 1)\). These angles not only simplify calculations but also enhance trigonometric understanding by offering reliable references during computations.

Sine Function

The sine function is a basic yet vital trigonometric function, which is central to understanding both theoretical and applied mathematics. Defined by the y-coordinate of a point on the unit circle, it translates an angle measure into a ratio.To comprehend the sine function:

• Imagine an angle originating from the positive x-axis, sweeping counterclockwise around the unit circle.
• The sine value of the angle corresponds to the vertical coordinate (y-value) of the point of intersection on the circle.
• Because of the unit circle, the maximum and minimum values for the sine function are 1 and -1 respectively.

Let's consider the special angle \( \frac{\pi}{2} \). At this angle, which equates to 90 degrees, the point on the unit circle hits \((0, 1)\). Therefore, the sine value equates to 1, demonstrating how straightforward trigonometric evaluation becomes with the unit circle.