Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161당신이 사용하는 게 아니라요?교과서 변경

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

문제 2.3.40

3장, 문제 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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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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