Textbook Question
Find the exact value of each expression. See Example 3. sin 1305°
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3. Unit Circle
Reference Angles
Problem 80
Textbook Question
Suppose θ is in the interval (90°, 180°). Find the sign of each of the following. cot(θ + 180°)
Verified step by step guidance1
Recall the definition of the cotangent function: \(\cot \alpha = \frac{\cos \alpha}{\sin \alpha}\).
Use the periodicity property of cotangent: \(\cot(\theta + 180^\circ) = \cot \theta\) because cotangent has a period of \(180^\circ\).
Since \(\theta\) is in the interval \((90^\circ, 180^\circ)\), determine the signs of \(\sin \theta\) and \(\cos \theta\) in this interval. In the second quadrant, \(\sin \theta > 0\) and \(\cos \theta < 0\).
Evaluate the sign of \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Since numerator is negative and denominator is positive, \(\cot \theta\) is negative in this interval.
Therefore, the sign of \(\cot(\theta + 180^\circ)\) is the same as the sign of \(\cot \theta\), which is negative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cotangent Function and Its Sign
Cotangent is the ratio of cosine to sine (cot θ = cos θ / sin θ). Its sign depends on the signs of sine and cosine in the given angle's quadrant. Understanding how cotangent behaves in different quadrants helps determine its sign.
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Angle Addition and Periodicity of Trigonometric Functions
Adding 180° to an angle shifts it by half a full rotation, affecting the signs of sine and cosine. Since cotangent has a period of 180°, cot(θ + 180°) = cot θ, meaning the function repeats every 180°.
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Quadrants and Sign of Trigonometric Functions
The interval (90°, 180°) places θ in the second quadrant, where sine is positive and cosine is negative. Knowing the signs of sine and cosine in each quadrant is essential to determine the sign of cotangent and related expressions.
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