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. Recognize that \( \cot(\alpha) = \tan(90^\circ - \alpha) \). Therefore, set \( 5\theta + 2^\circ = 90^\circ - (2\theta + 4^\circ) \).
2. Simplify the equation: \( 5\theta + 2^\circ = 90^\circ - 2\theta - 4^\circ \).
3. Combine like terms: \( 5\theta + 2^\circ + 2\theta + 4^\circ = 90^\circ \).
4. Simplify further: \( 7\theta + 6^\circ = 90^\circ \).
5. Solve for \( \theta \) by isolating it: \( 7\theta = 90^\circ - 6^\circ \).

Key Concepts

Cotangent and Tangent Functions

The cotangent function, cot(θ), is the reciprocal of the tangent function, tan(θ). This means that cot(θ) = 1/tan(θ). Understanding the relationship between these two functions is crucial for solving equations involving them, as it allows for the transformation of one function into another, facilitating the simplification of trigonometric equations.

Angle Addition Formulas

Angle addition formulas are essential for simplifying expressions involving sums of angles. For example, the tangent of a sum can be expressed as tan(A + B) = (tan A + tan B) / (1 - tan A tan B). In this problem, recognizing how to apply these formulas to the angles in the cotangent and tangent functions will help in finding solutions for the given equation.

Acute Angles in Trigonometry

Acute angles are angles that measure less than 90 degrees. In trigonometry, the values of sine, cosine, tangent, cotangent, and other functions are positive for acute angles. This property is important when solving trigonometric equations, as it restricts the possible solutions to those that fall within the first quadrant, ensuring that the angles involved remain acute.