In Exercises 49–59, find the exact value of each expression. Do not use a calculator.sin 22𝜋3

Verified step by step guidance

1. Recognize that the angle given is in radians: \( \frac{22\pi}{3} \).
2. Convert the angle to a more manageable form by finding its coterminal angle within the interval \([0, 2\pi)\).
3. To find a coterminal angle, subtract \(2\pi\) from \(\frac{22\pi}{3}\) until the angle is within \([0, 2\pi)\).
4. Calculate \(\frac{22\pi}{3} - 2\pi\) repeatedly until the angle is within the desired range.
5. Once the angle is within \([0, 2\pi)\), use the unit circle to find the exact value of \(\sin\) for that angle.

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. Angles measured in radians correspond to points on the unit circle, allowing for the determination of exact values for trigonometric functions at specific angles.

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 and its values at key angles (like 0, π/2, π, etc.) is crucial for finding exact values of sine at various angles.

Angle Reduction

Angle reduction involves simplifying an angle to an equivalent angle within a standard range, typically between 0 and 2π. For example, to find sin(22π/3), one can reduce this angle by subtracting multiples of 2π until it falls within the standard range. This technique is essential for calculating trigonometric values without a calculator, as it allows for easier reference to known values on the unit circle.