Concept Check Find a solution for each equation. tan (3θ ― 4°) = 1 / [cot(5θ ― 8°)]

Verified step by step guidance

1. Recognize that \( \tan(3\theta - 4^\circ) = \frac{1}{\cot(5\theta - 8^\circ)} \) can be rewritten using the identity \( \cot(x) = \frac{1}{\tan(x)} \).
2. Rewrite the equation as \( \tan(3\theta - 4^\circ) = \tan(5\theta - 8^\circ) \).
3. Since the tangent function is periodic with a period of \(180^\circ\), set up the equation \( 3\theta - 4^\circ = 5\theta - 8^\circ + k \times 180^\circ \), where \( k \) is an integer.
4. Solve for \( \theta \) by isolating it on one side of the equation: \( 3\theta - 5\theta = -8^\circ + 4^\circ + k \times 180^\circ \).
5. Simplify the equation to find \( \theta \) in terms of \( k \): \( -2\theta = -4^\circ + k \times 180^\circ \).

Key Concepts

Tangent and Cotangent Functions

The tangent function, tan(θ), is defined as the ratio of the opposite side to the adjacent side in a right triangle. Cotangent, cot(θ), is the reciprocal of tangent, expressed as cot(θ) = 1/tan(θ). Understanding these functions is crucial for solving equations involving angles, as they relate to the properties of triangles and the unit circle.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, reciprocal identities, and co-function identities. These identities can simplify complex trigonometric equations, making it easier to find solutions.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angle(s) that satisfy the equation. This often requires using algebraic manipulation, applying identities, and understanding the periodic nature of trigonometric functions. Solutions can be expressed in general form, accounting for the periodicity of the functions involved.