Ch. 6 - Inverse Circular Functions and Trigonometric Equations

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

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problema 6.2.33

Capitolo 7, Problema 6.2.33

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
csc² θ ―2 cot θ = 0

Guida verificata passo dopo passo

Recall the Pythagorean identity relating cosecant and cotangent: \(\csc^{2} \theta = 1 + \cot^{2} \theta\).

Substitute \(\csc^{2} \theta\) in the equation \(\csc^{2} \theta - 2 \cot \theta = 0\) with \(1 + \cot^{2} \theta\) to get \(1 + \cot^{2} \theta - 2 \cot \theta = 0\).

Rewrite the equation as a quadratic in terms of \(\cot \theta\): \(\cot^{2} \theta - 2 \cot \theta + 1 = 0\).

Solve the quadratic equation for \(\cot \theta\). Since it is a perfect square, factor it as \((\cot \theta - 1)^2 = 0\), which gives \(\cot \theta = 1\).

Find all angles \(\theta\) in the interval \([0^\circ, 360^\circ)\) where \(\cot \theta = 1\). Use the definition \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and find the corresponding \(\theta\) values.

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Reciprocal and Quotient Identities

Understanding that cosecant (csc θ) is the reciprocal of sine (sin θ), and cotangent (cot θ) is the quotient of cosine over sine (cos θ / sin θ), is essential. These identities allow rewriting the equation in terms of sine and cosine, simplifying the solving process.

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Pythagorean Identity involving Cotangent and Cosecant

The identity csc² θ = 1 + cot² θ relates cosecant and cotangent functions. This identity helps transform the given equation into a quadratic form in cot θ, making it easier to solve for θ within the specified interval.

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Pythagorean Identities

Solving Trigonometric Equations within a Given Interval

After simplifying, solving for θ requires finding all solutions in [0°, 360°). This involves considering the periodicity of trigonometric functions and using inverse trigonometric functions to find exact or approximate angle measures.

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