Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problema 80

Capitolo 3, Problema 80

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

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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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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°.

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Introduction 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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Quadratic Formula

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