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

Verified step by step guidance

1. Recall the Pythagorean identity relating cosecant and cotangent: \(\csc^{2} \theta = 1 + \cot^{2} \theta\).
2. Substitute \(\csc^{2} \theta\) in the equation \(\csc^{2} \theta - 2 \cot \theta = 0\) with \(1 + \cot^{2} \theta\) to get \(1 + \cot^{2} \theta - 2 \cot \theta = 0\).
3. Rewrite the equation as a quadratic in terms of \(\cot \theta\): \(\cot^{2} \theta - 2 \cot \theta + 1 = 0\).
4. Solve the quadratic equation for \(\cot \theta\). Since it is a perfect square, factor it as \((\cot \theta - 1)^2 = 0\), which gives \(\cot \theta = 1\).
5. 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.

Key Concepts

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.

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.

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.