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

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Rewrite the given equation \(\tan \theta - \cot \theta = 0\) by expressing \(\cot \theta\) in terms of \(\tan \theta\). Recall that \(\cot \theta = \frac{1}{\tan \theta}\).

Substitute \(\cot \theta\) with \(\frac{1}{\tan \theta}\) to get the equation \(\tan \theta - \frac{1}{\tan \theta} = 0\).

Multiply both sides of the equation by \(\tan \theta\) (noting that \(\tan \theta \neq 0\)) to eliminate the fraction, resulting in \(\tan^2 \theta - 1 = 0\).

Rewrite the equation as \(\tan^2 \theta = 1\) and then take the square root of both sides to find \(\tan \theta = \pm 1\).

Find all angles \(\theta\) in the interval \([0^\circ, 360^\circ)\) where \(\tan \theta = 1\) or \(\tan \theta = -1\). Use the unit circle or tangent values to identify these angles.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent and Cotangent Functions

Tangent (tan θ) is the ratio of the sine to the cosine of an angle, while cotangent (cot θ) is its reciprocal, cosine over sine. Understanding their definitions and properties is essential to manipulate and solve equations involving these functions.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a given interval. This requires knowledge of function periodicity and how to handle reciprocal identities to find exact or approximate angle values.

Interval Notation and Angle Measurement

The interval [0°, 360°) specifies the domain for solutions, meaning all angles must be found within one full rotation of the unit circle. Understanding how to interpret and restrict solutions to this interval ensures correct and complete answers.

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