Concept Check Find a solution for each equation. sec(2θ + 6°) cos(5θ + 3°) = 1

Verified step by step guidance

1. Recall the definition of secant: \(\sec x = \frac{1}{\cos x}\). Rewrite the equation \(\sec(2\theta + 6^\circ) \cos(5\theta + 3^\circ) = 1\) as \(\frac{1}{\cos(2\theta + 6^\circ)} \cdot \cos(5\theta + 3^\circ) = 1\).
2. Multiply both sides of the equation by \(\cos(2\theta + 6^\circ)\) to eliminate the fraction, giving \(\cos(5\theta + 3^\circ) = \cos(2\theta + 6^\circ)\).
3. Use the cosine equation property: if \(\cos A = \cos B\), then \(A = B + 360^\circ k\) or \(A = -B + 360^\circ k\), where \(k\) is any integer.
4. Set up the two equations based on the property: 1) \(5\theta + 3^\circ = 2\theta + 6^\circ + 360^\circ k\) 2) \(5\theta + 3^\circ = - (2\theta + 6^\circ) + 360^\circ k\)
5. Solve each linear equation for \(\theta\) by isolating \(\theta\) and expressing the general solution in terms of \(k\).

Key Concepts

Reciprocal Trigonometric Functions

The secant function, sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). Understanding this relationship allows us to rewrite or simplify equations involving secant and cosine, which is essential for solving the given equation.

Trigonometric Equation Solving

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the domain. This often requires using identities, inverse functions, and considering periodicity to find general solutions.

Angle Sum and Multiple Angle Identities

Expressions like 2θ + 6° and 5θ + 3° involve linear combinations of the variable θ. Understanding how to handle these angles, including their periodic properties and how to solve equations with multiple angles, is crucial for finding all valid solutions.