Problem 7 - Precalculus: Functions and Graphs

Question

Find the exact value. (a) \(\sin (2 \pi / 3)\) (b) \(\sin (-5 \pi / 4)\)

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{3}}{2}\), (b) \(-\frac{\sqrt{2}}{2}\)

Step 1: Use the Unit Circle for Part (a)

The angle \( \frac{2\pi}{3} \) is in the second quadrant. Knowing this, \( \sin(\theta) \) for angles in the second quadrant is positive and equivalent to \( \sin(\pi - \theta) \). Specifically, \( \sin(\frac{2\pi}{3}) = \sin(\pi - \frac{\pi}{3}) = \sin(\frac{\pi}{3}) \).

Step 2: Calculate \( \sin(\frac{\pi}{3}) \)

From the unit circle, \( \sin(\frac{\pi}{3}) \) is known to be \( \frac{\sqrt{3}}{2} \). Hence, \( \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2} \).

Step 3: Determine the Quadrant for Part (b)

The angle \(-\frac{5\pi}{4}\) can be rewritten as a positive angle by adding \(2\pi\): \(-\frac{5\pi}{4} + 2\pi = \frac{3\pi}{4}\). This is in the second quadrant.

Step 4: Calculate \( \sin(\frac{3\pi}{4}) \)

In the second quadrant, \( \sin(\theta) \) is positive and \( \sin(\frac{3\pi}{4}) = \sin(\pi - \frac{\pi}{4}) = \sin(\frac{\pi}{4}) \). Thus, \( \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2} \).

Step 5: Adjust for the Negative Angle

For \( \sin(-\frac{5\pi}{4}) \), we need to account for the fact that sine is an odd function, meaning \( \sin(-\theta) = -\sin(\theta) \). Therefore, \( \sin(-\frac{5\pi}{4}) = -\sin(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \).

Key Concepts

Unit Circle

The Unit Circle is an essential tool in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The circle helps to define sine, cosine, and tangent values for different angles, allowing you to easily visualize and compute trigonometric values.

The Unit Circle is often represented in radians, where:

• An angle of 0 or \(2\pi\) radians corresponds to the positive x-axis.
• \(\frac{\pi}{2}\) radians is at the positive y-axis.
• \(\pi\) radians is on the negative x-axis.
• \(\frac{3\pi}{2}\) radians is on the negative y-axis.

Knowing these positions helps to determine the sine, cosine, and tangent of common angles. Specifically, the sine of an angle corresponds to the y-coordinate of its point on the unit circle.

Quadrants in Trigonometry

Trigonometry divides the coordinate plane into four quadrants, and knowing in which quadrant an angle lies helps determine the sign of trigonometric functions.

Here’s a breakdown:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, but cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, while cosine is positive.

For the angle \(\frac{2\pi}{3}\), it lies in Quadrant II, hence sine is positive. For the angle \(-\frac{5\pi}{4}\), when rewritten in terms of positive rotation as \(\frac{3\pi}{4}\), it also lies in Quadrant II, maintaining the positive sine before considering the negative angle property.

Properties of Sine Function

The sine function has several key properties that aid in calculating its values:

• **Range:** The sine function ranges between -1 and 1.
• **Periodicity:** The sine function is periodic with a period of \(2\pi\), meaning \(\sin(\theta + 2\pi) = \sin(\theta)\).
• **Parity:** Sine is an odd function, which means \(\sin(-\theta) = -\sin(\theta)\).

For example, calculating \(\sin(-\frac{5\pi}{4})\) is simplified by recognizing the parity property: it equals \(-\sin(\frac{3\pi}{4})\). Understanding sine's characteristics helps to efficiently solve trigonometric problems, especially when combined with knowledge of the unit circle and quadrant properties.