Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.3.40

Chapter 3, Problem 2.3.40

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2. cot θ = 0.21563481

Verified step by step guidance

Recall the definition of cotangent in terms of tangent: \(\cot \theta = \frac{1}{\tan \theta}\). This means \(\tan \theta = \frac{1}{\cot \theta}\).

Calculate \(\tan \theta\) by taking the reciprocal of the given cotangent value: \(\tan \theta = \frac{1}{0.21563481}\).

Use the inverse tangent function to find \(\theta\): \(\theta = \tan^{-1} \left( \frac{1}{0.21563481} \right)\).

Make sure your calculator is set to degree mode since the problem asks for the answer in degrees.

Evaluate the inverse tangent expression to find \(\theta\) in degrees, then round your answer to six decimal places as requested.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cotangent Function

The cotangent of an angle θ in a right triangle is the ratio of the adjacent side to the opposite side, or equivalently, cot θ = 1 / tan θ. It is the reciprocal of the tangent function and is defined for all angles where tan θ is not zero.

Inverse Trigonometric Functions

To find an angle given a trigonometric ratio, we use inverse functions. For cotangent, θ = arccot(value), which can be computed as θ = arctan(1 / value). This allows us to determine the angle corresponding to a specific cotangent value.

Angle Measurement and Interval Restrictions

The problem restricts θ to the interval [0°, 90°), meaning the angle must be in the first quadrant where all trigonometric ratios are positive. This constraint ensures a unique solution and affects how inverse functions are interpreted.

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Textbook Question

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.

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