In Exercises 39–42, express the given quantity in terms of sin x and cos x.


sin (2π − x)

Verified step by step guidance

1. Recognize that the expression involves the sine of a difference: sin(2π - x). This can be simplified using the sine subtraction formula: sin(a - b) = sin(a)cos(b) - cos(a)sin(b).
2. Substitute a = 2π and b = x into the formula: sin(2π - x) = sin(2π)cos(x) - cos(2π)sin(x).
3. Recall the values of sin(2π) and cos(2π). Since 2π is a full rotation in the unit circle, sin(2π) = 0 and cos(2π) = 1.
4. Substitute these values into the expression: sin(2π - x) = 0 * cos(x) - 1 * sin(x).
5. Simplify the expression: sin(2π - x) = -sin(x).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One important identity is the sine and cosine of complementary angles, which states that sin(π - x) = sin(x) and cos(π - x) = -cos(x). These identities are essential for simplifying expressions involving trigonometric functions.

Sine Function Properties

The sine function, denoted as sin(x), is a periodic function that represents the y-coordinate of a point on the unit circle corresponding to an angle x. It has specific properties, such as sin(2π - x) = sin(x), which reflects the symmetry of the sine function about the y-axis. Understanding these properties helps in transforming trigonometric expressions.

Angle Subtraction

Angle subtraction refers to the process of finding the sine or cosine of an angle expressed as the difference between two angles. In this case, sin(2π - x) can be analyzed using the periodic nature of the sine function, which allows us to express it in terms of sin(x) and cos(x). Recognizing how angles relate to one another is crucial for simplifying trigonometric expressions.