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Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblema 66
Capitolo 3, Problema 66

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. √3 cot θ = - —— 3

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Start by writing down the given equation: \(\cot \theta = -\frac{\sqrt{3}}{3}\).
Recall that \(\cot \theta = \frac{1}{\tan \theta}\), so rewrite the equation as \(\frac{1}{\tan \theta} = -\frac{\sqrt{3}}{3}\), which implies \(\tan \theta = -\frac{3}{\sqrt{3}}\).
Simplify \(\tan \theta = -\frac{3}{\sqrt{3}}\) to get \(\tan \theta = -\sqrt{3}\).
Determine the reference angle where \(\tan \theta = \sqrt{3}\). From the unit circle or common angles, \(\tan 60^\circ = \sqrt{3}\), so the reference angle is \(60^\circ\).
Since \(\tan \theta\) is negative, find all angles in the interval \([0^\circ, 360^\circ)\) where tangent is negative. Tangent is negative in the second and fourth quadrants, so the solutions are \(\theta = 180^\circ - 60^\circ\) and \(\theta = 360^\circ - 60^\circ\).

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Cotangent Function and Its Definition

Cotangent (cot θ) is the reciprocal of the tangent function, defined as cot θ = cos θ / sin θ. Understanding this relationship helps in converting cotangent values into sine and cosine ratios, which is essential for solving trigonometric equations.
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Introduction to Cotangent Graph

Solving Trigonometric Equations in a Given Interval

When solving for θ within [0°, 360°), it is important to find all angles that satisfy the equation. This involves considering the periodicity of trigonometric functions and identifying all solutions in the specified range, including those in different quadrants.
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Sign of Trigonometric Functions in Different Quadrants

The sign of cotangent depends on the signs of sine and cosine, which vary by quadrant. Cotangent is negative where sine and cosine have opposite signs, specifically in the second and fourth quadrants, guiding the selection of correct angle solutions.
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Introduction to Trigonometric Functions
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