Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
(2 tan θ) / (3 - tan² θ ) = 1

Verified step by step guidance

1. Start by letting \(t = \tan \theta\) to simplify the equation. The given equation is \(\frac{2 \tan \theta}{3 - \tan^2 \theta} = 1\), which becomes \(\frac{2t}{3 - t^2} = 1\).
2. Multiply both sides of the equation by the denominator \((3 - t^2)\) to clear the fraction: \$2t = 3 - t^2$.
3. Rearrange the equation to standard quadratic form: \(t^2 + 2t - 3 = 0\).
4. Solve the quadratic equation \(t^2 + 2t - 3 = 0\) using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=2\), and \(c=-3\).
5. Once you find the values of \(t = \tan \theta\), solve for \(\theta\) by taking the arctangent of each solution. Remember to find all solutions within the specified domain, using the periodicity of the tangent function, and express answers in both radians and degrees as required.

Key Concepts

Trigonometric Equations and Their Solutions

Trigonometric equations involve functions like sine, cosine, and tangent. Solving these requires isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain, often considering periodicity and principal values.

Tangent Function and Its Properties

The tangent function, tan(θ), is periodic with period π and undefined at odd multiples of π/2. Understanding its behavior, including its range and asymptotes, is essential for solving equations involving tan(θ) and for determining valid solutions.

Use of Exact and Approximate Solutions in Radians and Degrees

Exact solutions are expressed in terms of known angles or constants, while approximate solutions use decimal values rounded appropriately. Converting between radians and degrees and expressing answers within the least nonnegative angle measure ensures clarity and correctness.