Suppose θ is in the interval (90°, 180°). Find the sign of each of the following. sin(-θ)

Key Concepts

Understanding Quadrants

In trigonometry, the unit circle is divided into four quadrants, each corresponding to specific ranges of angles. The first quadrant (0° to 90°) has all positive sine and cosine values, while the second quadrant (90° to 180°) has a positive sine and a negative cosine. Knowing the quadrant in which an angle lies helps determine the signs of the trigonometric functions.

Properties of Sine Function

The sine function is an odd function, meaning that sin(-θ) = -sin(θ). This property is crucial for evaluating sine values for negative angles. Understanding this property allows us to relate the sine of a negative angle to the sine of its positive counterpart, which is essential for solving the given problem.

Angle Reference and Sign Determination

When dealing with angles in trigonometry, it's important to consider the reference angle and its position in the unit circle. For θ in the interval (90°, 180°), the reference angle is 180° - θ. Since sine is positive in the second quadrant, we can conclude that sin(θ) is positive, and thus sin(-θ) will be negative due to the odd function property.