Textbook Question
Find the exact value of each real number y. Do not use a calculator.
y = sin⁻¹ √2/2
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 6.55
Textbook Question
Use a calculator to approximate each value in decimal degrees.
θ = cot⁻¹ (-0.60724226)
Verified step by step guidance1
Understand that \( \cot^{-1}(x) \) is the inverse cotangent function, which gives the angle \( \theta \) whose cotangent is \( x \).
Recognize that the cotangent function is the reciprocal of the tangent function, so \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
Since \( \theta = \cot^{-1}(-0.60724226) \), we need to find the angle whose cotangent is \(-0.60724226\).
Use a calculator to find \( \theta \) by inputting \( \cot^{-1}(-0.60724226) \). Ensure the calculator is set to degree mode to get the answer in decimal degrees.
Interpret the result: The angle \( \theta \) will be in the range of \( (0, 180) \) degrees for the inverse cotangent function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Cotangent Function
The inverse cotangent function, denoted as cot⁻¹ or arccot, is used to find an angle whose cotangent is a given value. It is defined for all real numbers and returns angles in the range of 0 to π radians (or 0 to 180 degrees). Understanding this function is crucial for solving problems involving angles derived from cotangent values.
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Calculator Functions
Using a scientific calculator effectively is essential for approximating trigonometric values. Most calculators have specific functions for inverse trigonometric operations, including cotangent. Familiarity with how to input values and interpret results in decimal degrees is necessary for accurate calculations.
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Quadrants and Angle Signs
In trigonometry, the sign of the cotangent value indicates the quadrant in which the angle lies. Since cotangent is negative in the second and fourth quadrants, recognizing this helps determine the correct angle when using the inverse function. This understanding is vital for interpreting the results of cot⁻¹ correctly.
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