In Exercises 49–59, find the exact value of each expression. Do not use a calculator.sin(-𝜋/3)

Verified step by step guidance

1. Recognize that \( \sin(-\theta) = -\sin(\theta) \) due to the odd property of the sine function.
2. Substitute \( \theta = \frac{\pi}{3} \) into the identity: \( \sin(-\frac{\pi}{3}) = -\sin(\frac{\pi}{3}) \).
3. Recall the exact value of \( \sin(\frac{\pi}{3}) \), which is \( \frac{\sqrt{3}}{2} \).
4. Apply the negative sign from the identity: \( -\sin(\frac{\pi}{3}) = -\frac{\sqrt{3}}{2} \).
5. Conclude that \( \sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2} \).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured from the positive x-axis, allowing for easy calculation of trigonometric values.

Sine Function

The sine function, denoted as sin(θ), represents the y-coordinate of a point on the unit circle corresponding to an angle θ. It is periodic with a period of 2π, meaning that sin(θ) = sin(θ + 2nπ) for any integer n. Understanding the sine function's behavior, including its values for common angles, is crucial for evaluating expressions like sin(-π/3).

Negative Angles

In trigonometry, negative angles indicate a clockwise rotation from the positive x-axis. The sine function is an odd function, which means that sin(-θ) = -sin(θ). This property simplifies the evaluation of sine for negative angles, allowing us to find sin(-π/3) by using the positive angle equivalent, leading to the conclusion that sin(-π/3) = -sin(π/3).