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

Verified step by step guidance

1. Recall the identity relating tangent and cotangent: \(\tan x = \frac{1}{\cot x}\). This means the equation \(\tan(3\theta - 4^\circ) = \frac{1}{\cot(5\theta - 8^\circ)}\) can be rewritten using this identity.
2. Rewrite the right side using the identity: \(\frac{1}{\cot(5\theta - 8^\circ)} = \tan(5\theta - 8^\circ)\). So the equation becomes \(\tan(3\theta - 4^\circ) = \tan(5\theta - 8^\circ)\).
3. Use the property that if \(\tan A = \tan B\), then \(A = B + k \times 180^\circ\), where \(k\) is any integer. Set up the equation: \(3\theta - 4^\circ = 5\theta - 8^\circ + k \times 180^\circ\).
4. Solve the equation for \(\theta\): Rearrange terms to isolate \(\theta\) on one side, which gives \(3\theta - 5\theta = -8^\circ + 4^\circ + k \times 180^\circ\), simplifying to \(-2\theta = -4^\circ + k \times 180^\circ\).
5. Divide both sides by \(-2\) to find \(\theta\): \(\theta = \frac{4^\circ - k \times 180^\circ}{2}\). This expression gives the general solution for \(\theta\) depending on integer values of \(k\).

Key Concepts

Relationship Between Tangent and Cotangent

Tangent and cotangent are reciprocal trigonometric functions, meaning tan(x) = 1/cot(x) and cot(x) = 1/tan(x). Recognizing this relationship allows simplification of equations involving both functions by converting one into the other.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the domain. Since trigonometric functions are periodic, solutions often include general forms with added multiples of the function's period.

Angle Manipulation and Equation Setup

Understanding how to manipulate angles inside trigonometric functions, such as linear expressions like 3θ - 4°, is essential. Setting up the equation correctly by equating angles or their trigonometric values helps in finding the variable θ.