Textbook Question
Use a calculator to approximate each value in decimal degrees.
θ = csc⁻¹ 1.9422833
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 6.57
Textbook Question
Use a calculator to approximate each value in decimal degrees.
θ = tan⁻¹ (-7.7828641)
Verified step by step guidance1
Identify the trigonometric function involved: In this case, it's the inverse tangent function, denoted as \( \tan^{-1} \).
Understand that \( \tan^{-1} \) is used to find an angle \( \theta \) whose tangent is the given value, which is \(-7.7828641\) in this problem.
Use a calculator to compute \( \theta = \tan^{-1}(-7.7828641) \). Ensure your calculator is set to degree mode to get the result in decimal degrees.
Recognize that the result will be an angle in the fourth quadrant because the tangent of a negative value corresponds to an angle between \(-90^\circ\) and \(0^\circ\).
Interpret the calculator's output as the angle \( \theta \) in decimal degrees, which is the solution to the problem.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Tangent Function
The inverse tangent function, denoted as tan⁻¹ or arctan, is used to find an angle whose tangent is a given number. It is particularly useful in trigonometry for determining angles in right triangles when the lengths of the opposite and adjacent sides are known. The output of the inverse tangent function is typically in radians or degrees, depending on the calculator settings.
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Calculator Settings
When using a calculator for trigonometric functions, it is crucial to ensure that the calculator is set to the correct mode: degrees or radians. This setting affects the output of functions like tan⁻¹, as the same input can yield different angle measures based on the mode. For this question, the output should be in decimal degrees, so the calculator must be set accordingly.
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Negative Tangent Values
The tangent function can yield negative values, which correspond to angles in specific quadrants of the unit circle. For example, a negative tangent value indicates that the angle is in the second or fourth quadrant. Understanding the behavior of the tangent function and its inverse helps in interpreting the results correctly, especially when dealing with angles that may not be immediately intuitive.
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