Find the exact value of each expression.
sin 255°

Verified step by step guidance

1. Step 1: Recognize that 255° is not a standard angle, so we need to express it in terms of known angles.
2. Step 2: Use the angle subtraction identity for sine: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).
3. Step 3: Express 255° as 270° - 15°, which are angles we can work with.
4. Step 4: Apply the identity: \( \sin(255°) = \sin(270° - 15°) = \sin 270° \cos 15° - \cos 270° \sin 15° \).
5. Step 5: Substitute the known values: \( \sin 270° = -1 \), \( \cos 270° = 0 \), and use the values for \( \cos 15° \) and \( \sin 15° \) from trigonometric tables or identities.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the sine, cosine, and tangent functions based on the coordinates of points on a circle with a radius of one. Each angle corresponds to a point on the circle, where the x-coordinate represents the cosine and the y-coordinate represents the sine of that angle. Understanding the unit circle allows for the determination of exact values for trigonometric functions at various angles.

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. For angles greater than 180°, like 255°, the reference angle helps simplify the calculation of trigonometric values by relating them to angles in the first quadrant. In this case, the reference angle for 255° is 255° - 180° = 75°, which can be used to find the sine value.

Sine Function Properties

The sine function is periodic and has specific properties based on the quadrant in which the angle lies. For angles in the third quadrant, such as 255°, the sine value is negative. Knowing that sin(θ) = -sin(reference angle) for angles in the third quadrant allows us to determine that sin(255°) = -sin(75°), leading to the exact value calculation.