Problem 25 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \sin \frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

\( \sin \frac{3\pi}{2} = -1 \)

Step 1: Understanding the Angle

The angle given is \( \frac{3\pi}{2} \). This angle is in radians. To understand its location, we convert it to degrees. We know \( \pi = 180^\circ \). Thus, \( \frac{3\pi}{2} = \frac{3 \times 180}{2} = 270^\circ \). The angle of \( 270^\circ \) is located on the negative y-axis of the unit circle.

Step 2: Identifying the Sine Value

The sine function represents the y-coordinate of a point on the unit circle. At \( 270^\circ \), the point is at the bottom of the unit circle on the negative y-axis. The coordinates at \( 270^\circ \) are \((0, -1)\). Therefore, \( \sin \frac{3\pi}{2} = -1 \).

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 centered at the origin of a coordinate plane.

• The circle's circumference is defined by all points that satisfy the equation \( x^2 + y^2 = 1 \).
• Angles in the unit circle are measured from the positive x-axis and can be expressed in radians or degrees.
• The circle's importance lies in the ease with which it allows us to define trigonometric functions like sine, cosine, and tangent based on the coordinates of points along its circumference.

Each point on the unit circle is represented as \((x, y)\), where \(x\) and \(y\) are the cosine and sine values of the corresponding angle, respectively. This relationship allows us to find trigonometric values by imagining or calculating the central angle's position along the unit circle. For example, an angle of \(270^\circ\) or \(\frac{3\pi}{2}\) radians on the unit circle corresponds to the coordinates \((0, -1)\). This reflects directly in the value of the sine function since the sine of an angle is represented by the y-coordinate on the unit circle.

Angle Conversion: Radians and Degrees

Angles in trigonometry can be expressed in two primary units: degrees and radians.

• Degrees: One complete rotation around a circle is \(360^\circ\).
• Radians: Another way to measure angles, where one complete revolution equals \(2\pi\) radians.

To convert between degrees and radians, we use the relationship \(\pi\) radians = \(180^\circ\). Hence: - To convert degrees to radians, use the formula \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).- Conversely, to convert radians to degrees, use \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\).Consider an example: converting \(\frac{3\pi}{2}\) radians to degrees. By applying the formula, \(\frac{3\pi}{2} \times \frac{180}{\pi} = 270^\circ\). This conversion helps us visualize where the angle is located on the unit circle, confirming that \(270^\circ\) lies on the negative y-axis.

Exploring the Sine Function

The sine function is a key trigonometric function that provides the y-coordinate of a point on the unit circle.

• The sine of an angle \(\theta\) is equal to the y-value of the unit circle coordinate for that angle.
• This function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians (or \(360^\circ\)).
• Sine values range between -1 and 1.

For an angle like \(\frac{3\pi}{2}\), which is equal to \(270^\circ\), we find its sine by locating the corresponding point on the unit circle. At this position, the point is \((0, -1)\), giving a sine value of \(-1\). This value indicates the lowest point on the sine wave cycle, emphasizing its symmetry and periodic nature. Such knowledge of sine's behavior and unit circle association provides us a solid foundation in analyzing periodic phenomena and waves in the realm of mathematics and beyond.