Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. cot(5θ + 2°) = tan(2θ + 4°)

Verified step by step guidance

1. Recall the trigonometric identity that relates cotangent and tangent: \(\cot x = \tan(90^\circ - x)\). This will help us rewrite the equation in a more comparable form.
2. Rewrite the given equation \(\cot(5\theta + 2^\circ) = \tan(2\theta + 4^\circ)\) using the identity: \(\tan(90^\circ - (5\theta + 2^\circ)) = \tan(2\theta + 4^\circ)\).
3. Since \(\tan A = \tan B\) implies that \(A = B + k \times 180^\circ\) for any integer \(k\), set up the equation: \(90^\circ - (5\theta + 2^\circ) = 2\theta + 4^\circ + k \times 180^\circ\).
4. Simplify the equation to isolate \(\theta\): \(90^\circ - 5\theta - 2^\circ = 2\theta + 4^\circ + k \times 180^\circ\) which simplifies to \(88^\circ - 5\theta = 2\theta + 4^\circ + k \times 180^\circ\).
5. Solve for \(\theta\) by bringing all \(\theta\) terms to one side and constants to the other: \(88^\circ - 4^\circ - k \times 180^\circ = 2\theta + 5\theta\) which simplifies to \(84^\circ - k \times 180^\circ = 7\theta\). Then, \(\theta = \frac{84^\circ - k \times 180^\circ}{7}\). Choose \(k\) such that \(\theta\) is an acute angle (between \(0^\circ\) and \(90^\circ\)).

Key Concepts

Relationship Between Cotangent and Tangent

Cotangent and tangent are reciprocal trigonometric functions related by cot(x) = tan(90° - x). This identity allows transforming cotangent expressions into tangent ones, facilitating equation solving by comparing angles or using complementary angle properties.

Solving Trigonometric Equations Involving Angles

Solving equations like cot(5θ + 2°) = tan(2θ + 4°) involves using trigonometric identities and algebraic manipulation to isolate θ. Recognizing angle relationships and applying inverse functions help find angle measures that satisfy the equation.

Acute Angle Assumption and Its Implications

Assuming all angles are acute (between 0° and 90°) restricts possible solutions, ensuring the angles lie within the first quadrant. This constraint simplifies the solution process by limiting the domain and avoiding extraneous or non-physical solutions.