Textbook Question
Concept Check Work each problem. Without using a calculator, determine which of the following numbers is closest to sin 115°: -0.9, -0.1, 0, 0.1, or 0.9.
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Ch. 2 - Acute Angles and Right Triangles
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 3, Problem 80
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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Introduction to Cotangent Graph
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°.
Recommended video:
6:04Introduction to Trigonometric Functions
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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6:36Quadratic Formula
Related Practice
Textbook Question
Find a formula for the area of each figure in terms of s.
Textbook Question
Suppose θ is in the interval (90°, 180°). Find the sign of each of the following. sec(θ + 180°)
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Textbook Question
Concept Check Work each problem. For what angles θ between 0° and 360° is cos θ = sin θ true?
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Textbook Question
Find a formula for the area of each figure in terms of s.
Textbook Question
Suppose θ is in the interval (90°, 180°). Find the sign of each of the following. sin(-θ)
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