Problem 3 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value is 1, since \( \sin \frac{\pi}{2} = 1 \).

Step 1: Identify the Angle

The first step is to recognize that the angle given is \( \frac{\pi}{2} \). This angle is measured in radians and is a commonly referenced angle on the unit circle.

Step 2: Locate the Angle on the Unit Circle

Next, place \( \frac{\pi}{2} \) on the unit circle. This angle is found at the top of the circle, where the unit circle intersects the y-axis.

Step 3: Determine the Sine of the Angle

On the unit circle, the sine of an angle corresponds to the y-coordinate of its terminal point. At \( \frac{\pi}{2} \), the coordinates are (0, 1).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is a powerful tool in understanding trigonometric functions because it allows you to visualize angles and their corresponding function values. When you plot an angle on the unit circle, you're essentially drawing a line from the center to the circumference, creating what's called a terminal side. For any angle, the position where the terminal side meets the circle has coordinates

• Cosine of the angle - the x-coordinate
• Sine of the angle - the y-coordinate

Thus, the unit circle not only helps determine the values of trigonometric functions but also reinforces the concept of using both radians and degrees for angle measurement.
This makes it much easier to see how angles relate to each other and to anticipate their sine and cosine values without needing a calculator.

Sine Function

The sine function is one of the main trigonometric functions and is fundamental in mathematics. It relates the angle of a right triangle to the ratio of the length of the opposite side over the hypotenuse. When using the unit circle, the sine of an angle is found as the y-coordinate of its corresponding point on the circle.
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Angle in Radians

Angles can be measured in degrees or radians, and radians are used extensively in advanced mathematics because they provide a direct relationship to the arc length of a circle. One complete revolution around the unit circle corresponds to an angle of 2π radians or 360 degrees. When using radians, the angle is expressed in terms of π, which makes it easier to compute trigonometric functions. For instance, the angle π/2 is one-quarter the way around the circle.
This means:

• π/2 is equivalent to 90 degrees, which is at the vertical top of the circle.
• At this position, the sine is 1 because the y-coordinate reflects the maximum vertical height of the circle.

Understanding radians and their placement on the unit circle is essential for solving problems in trigonometry and calculus.