Ch. 2 - Acute Angles and Right Triangles

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Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problema 40

Capitolo 3, Problema 40

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

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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 \(\frac{1}{\sin(\beta + 40^\circ)} = \frac{1}{\cos(\beta - 20^\circ)}\).

Cross-multiply to eliminate the fractions, giving \(\cos(\beta - 20^\circ) = \sin(\beta + 40^\circ)\).

Use the co-function identity \(\sin \theta = \cos(90^\circ - \theta)\) to rewrite the right side: \(\sin(\beta + 40^\circ) = \cos(90^\circ - (\beta + 40^\circ)) = \cos(50^\circ - \beta)\).

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.

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 \(90^\circ\)).

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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.

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Introduction to Trigonometric Functions

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.

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How to Solve Linear Trigonometric Equations

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.

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Finding the Domain of an Equation

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