Ch. 2 - Acute Angles and Right Triangles

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Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problema 36

Capitolo 3, Problema 36

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. cot(5θ + 2°) = tan(2θ + 4°)

Guida verificata passo dopo passo

Recall the trigonometric identity that relates cotangent and tangent: \(\cot x = \tan(90^\circ - x)\). This will help us rewrite the equation in a more comparable form.

Rewrite the given equation \(\cot(5\theta + 2^\circ) = \tan(2\theta + 4^\circ)\) using the identity: \(\tan(90^\circ - (5\theta + 2^\circ)) = \tan(2\theta + 4^\circ)\).

Since \(\tan A = \tan B\) implies that \(A = B + k \times 180^\circ\) for any integer \(k\), set up the equation: \(90^\circ - (5\theta + 2^\circ) = 2\theta + 4^\circ + k \times 180^\circ\).

Simplify the equation to isolate \(\theta\): \(90^\circ - 5\theta - 2^\circ = 2\theta + 4^\circ + k \times 180^\circ\) which simplifies to \(88^\circ - 5\theta = 2\theta + 4^\circ + k \times 180^\circ\).

Solve for \(\theta\) by bringing all \(\theta\) terms to one side and constants to the other: \(88^\circ - 4^\circ - k \times 180^\circ = 2\theta + 5\theta\) which simplifies to \(84^\circ - k \times 180^\circ = 7\theta\). Then, \(\theta = \frac{84^\circ - k \times 180^\circ}{7}\). Choose \(k\) such that \(\theta\) is an acute angle (between \(0^\circ\) and \(90^\circ\)).

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Relationship Between Cotangent and Tangent

Cotangent and tangent are reciprocal trigonometric functions related by cot(x) = tan(90° - x). This identity allows transforming cotangent expressions into tangent ones, facilitating equation solving by comparing angles or using complementary angle properties.

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Introduction to Cotangent Graph

Solving Trigonometric Equations Involving Angles

Solving equations like cot(5θ + 2°) = tan(2θ + 4°) involves using trigonometric identities and algebraic manipulation to isolate θ. Recognizing angle relationships and applying inverse functions help find angle measures that satisfy the equation.

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How to Solve Linear Trigonometric Equations

Acute Angle Assumption and Its Implications

Assuming all angles are acute (between 0° and 90°) restricts possible solutions, ensuring the angles lie within the first quadrant. This constraint simplifies the solution process by limiting the domain and avoiding extraneous or non-physical solutions.

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Drawing Angles in Standard Position

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