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.
sin θ = 0.84802194
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1. Measuring Angles
Angles in Standard Position
1:54 minutesProblem 2.3.40
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. cot θ = 0.21563481
Verified step by step guidance1
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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Video duration:
1mKey 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.
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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.
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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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