Concept CheckSuppose that 90° < θ < 180° .Find the sign of each function value. cot (θ + 180°)

Verified step by step guidance

1. Recognize that \( \theta \) is in the second quadrant since \( 90^\circ < \theta < 180^\circ \).
2. Recall that the cotangent function, \( \cot(\theta) \), is negative in the second quadrant because \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) and \( \cos(\theta) \) is negative while \( \sin(\theta) \) is positive in this quadrant.
3. Use the identity \( \cot(\theta + 180^\circ) = \cot(\theta) \) because adding 180° to an angle results in the same cotangent value but in the opposite direction on the unit circle.
4. Since \( \cot(\theta) \) is negative in the second quadrant, \( \cot(\theta + 180^\circ) \) will have the same sign as \( \cot(\theta) \) in the fourth quadrant, where cotangent is positive.
5. Conclude that \( \cot(\theta + 180^\circ) \) is positive.

Key Concepts

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function. It is defined as cot(θ) = cos(θ) / sin(θ). Understanding the behavior of cotangent in different quadrants is essential, as its sign depends on the signs of sine and cosine in those quadrants.

Angle Addition

The angle addition formula allows us to find the trigonometric function of a sum of angles. For cotangent, cot(θ + 180°) can be simplified using the identity cot(θ + 180°) = cot(θ). This property is crucial for determining the sign of cotangent in the specified range of θ.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each with distinct signs for sine and cosine. In the second quadrant (90° < θ < 180°), sine is positive and cosine is negative. This understanding is vital for determining the sign of cotangent, as it relies on the signs of sine and cosine.