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

Verified step by step guidance

1. Recognize that \( \theta \) is in the second quadrant, where angles range from 90° to 180°.
2. Recall that the secant function, \( \sec(\theta) \), is the reciprocal of the cosine function, \( \cos(\theta) \).
3. Understand that \( \theta + 180° \) moves the angle to the third quadrant, where angles range from 180° to 270°.
4. In the third quadrant, the cosine of an angle is negative, so \( \cos(\theta + 180°) < 0 \).
5. Since \( \sec(\theta + 180°) = \frac{1}{\cos(\theta + 180°)} \), and \( \cos(\theta + 180°) \) is negative, \( \sec(\theta + 180°) \) is also negative.

Key Concepts

Trigonometric Functions and Their Signs

Trigonometric functions such as secant, sine, and cosine have specific signs in different quadrants of the unit circle. In the interval (90°, 180°), which corresponds to the second quadrant, the sine function is positive while the cosine function is negative. Since secant is the reciprocal of cosine, its sign will also be negative in this quadrant.

Angle Addition in Trigonometry

The angle addition formula allows us to find the value of trigonometric functions for the sum of two angles. For sec(θ + 180°), we can use the property that sec(θ + 180°) = sec(θ) because the secant function has a periodicity of 360°. This means that the sign of sec(θ + 180°) will be the same as that of sec(θ).

Periodicity of Trigonometric Functions

Trigonometric functions exhibit periodic behavior, meaning their values repeat at regular intervals. For secant, the period is 360°, which implies that sec(θ + 180°) is equivalent to -sec(θ). Understanding this periodicity is crucial for determining the sign of sec(θ + 180°) based on the sign of sec(θ) in the specified interval.