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. Recall the definition of the sine function for negative angles: \(\sin(-\theta) = -\sin(\theta)\).
2. Identify the positive angle corresponding to the given negative angle: here, \(\theta = \frac{\pi}{3}\).
3. Use the identity to rewrite the expression: \(\sin\left(-\frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)\).
4. Recall the exact value of \(\sin\left(\frac{\pi}{3}\right)\), which is \(\frac{\sqrt{3}}{2}\).
5. Substitute this value back into the expression to get \(\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\).

Key Concepts

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles are measured in radians, where 𝜋 radians equal 180 degrees. Understanding the position of angles like -𝜋/3 on the unit circle helps determine the sine value based on the y-coordinate of the corresponding point.

Sine Function and Its Properties

The sine function relates an angle to the y-coordinate of a point on the unit circle. It is an odd function, meaning sin(-θ) = -sin(θ). This property allows simplification of sine values for negative angles by converting them to positive angles and then negating the result.

Exact Values of Special Angles

Certain angles, such as 𝜋/3, 𝜋/4, and 𝜋/6, have well-known exact sine values derived from special right triangles. For 𝜋/3, sin(𝜋/3) = √3/2. Using these exact values avoids approximation and calculator use, enabling precise answers in trigonometric problems.