Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. csc(β + 40°) = sec(β - 20°)

Verified step by step guidance

1. Recall the definitions of the cosecant and secant functions: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\). Rewrite the given equation \(\csc(\beta + 40^\circ) = \sec(\beta - 20^\circ)\) as \(\frac{1}{\sin(\beta + 40^\circ)} = \frac{1}{\cos(\beta - 20^\circ)}\).
2. Cross-multiply to eliminate the fractions, giving \(\cos(\beta - 20^\circ) = \sin(\beta + 40^\circ)\).
3. Use the co-function identity \(\sin \theta = \cos(90^\circ - \theta)\) to rewrite the right side: \(\sin(\beta + 40^\circ) = \cos(90^\circ - (\beta + 40^\circ)) = \cos(50^\circ - \beta)\).
4. Set the two cosine expressions equal: \(\cos(\beta - 20^\circ) = \cos(50^\circ - \beta)\). Recall that if \(\cos A = \cos B\), then either \(A = B\) or \(A = 360^\circ - B\) (or in degrees, \(A = B + 360^\circ k\) or \(A = -B + 360^\circ k\) for integer \(k\)). Since angles are acute, focus on the principal solutions.
5. Solve the equations \(\beta - 20^\circ = 50^\circ - \beta\) and \(\beta - 20^\circ = -(50^\circ - \beta)\) for \(\beta\), then check which solutions are acute angles (between \(0^\circ\) and \(90^\circ\)).

Key Concepts

Reciprocal Trigonometric Functions

Cosecant (csc) and secant (sec) are reciprocal functions of sine and cosine, respectively. Specifically, csc(θ) = 1/sin(θ) and sec(θ) = 1/cos(θ). Understanding these relationships allows rewriting the equation in terms of sine and cosine for easier manipulation.

Trigonometric Equation Solving

Solving trigonometric equations often involves rewriting expressions, using identities, and isolating the variable. Here, equating csc(β + 40°) to sec(β - 20°) requires converting to sine and cosine, then finding β that satisfies the resulting equation within the given domain.

Domain Restrictions and Acute Angles

The problem specifies that all angles are acute, meaning they lie between 0° and 90°. This restriction limits possible solutions and helps in selecting the correct angle values after solving the equation, ensuring the solution is valid within the given context.