Domanda del libro di testo
Determine whether each statement is possible or impossible. See Example 4. cot θ = ―6
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Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- 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
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 67
Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find csc θ , given that cot θ = ―1/2 and θ is in quadrant IV.
Guida verificata passo dopo passo1
Recall the definitions of the trigonometric functions involved: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Since \( \cot \theta = -\frac{1}{2} \), we can write \( \frac{\cos \theta}{\sin \theta} = -\frac{1}{2} \).
Express \( \cos \theta \) in terms of \( \sin \theta \) using the given cotangent: \( \cos \theta = -\frac{1}{2} \sin \theta \).
Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to substitute \( \cos \theta \) and solve for \( \sin \theta \). This gives: \( \sin^2 \theta + \left(-\frac{1}{2} \sin \theta\right)^2 = 1 \).
Simplify the equation to find \( \sin^2 \theta \), then take the square root to find \( \sin \theta \). Remember to consider the sign of \( \sin \theta \) based on the quadrant: since \( \theta \) is in quadrant IV, \( \sin \theta \) is negative.
Finally, find \( \csc \theta = \frac{1}{\sin \theta} \) and rationalize the denominator if necessary to express the answer in simplest form.
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable where both sides are defined. Key identities like cot²θ + 1 = csc²θ allow us to relate cotangent and cosecant, enabling the calculation of one function given the other.
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Percorso guidato
5:32
Fundamental Trigonometric Identities
Sign of Trigonometric Functions in Quadrants
The sign of trigonometric functions depends on the quadrant in which the angle lies. In quadrant IV, sine and cosecant are negative, while cosine and secant are positive. This information is crucial to determine the correct sign of the solution when using identities.
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Percorso guidato
6:36Quadratic Formula
Rationalizing Denominators
Rationalizing denominators involves eliminating radicals or complex expressions from the denominator of a fraction. This process simplifies the expression and is often required for final answers in trigonometry problems to maintain standard form.
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Percorso guidato
2:58Rationalizing Denominators
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