Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

Trigonometry

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

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Problem 6.65

Textbook Question

Use a calculator to approximate each real number value. (Be sure the calculator is in radian mode.)
y = cot⁻¹ (―0.92170128)

Verified step by step guidance

Ensure your calculator is in radian mode. This is crucial because trigonometric functions can be calculated in either degrees or radians, and the problem specifies radian mode.

Understand that \( \cot^{-1}(x) \) is the inverse cotangent function, which gives the angle whose cotangent is \( x \).

Recognize that \( \cot^{-1}(x) \) is equivalent to \( \tan^{-1}(\frac{1}{x}) \) because cotangent is the reciprocal of tangent.

Calculate \( \tan^{-1}(\frac{1}{-0.92170128}) \) using your calculator. This will give you the angle in radians.

Interpret the result from the calculator as the angle \( y \) in radians, which is the solution to \( y = \cot^{-1}(-0.92170128) \).

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Key Concepts

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

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cotangent inverse (cot⁻¹), are used to find the angle whose cotangent is a given number. These functions are essential for solving equations where the angle is unknown. The output of these functions is typically restricted to a specific range to ensure that each input corresponds to one unique output.

Cotangent Function

The cotangent function is defined as the ratio of the adjacent side to the opposite side in a right triangle, or as the reciprocal of the tangent function. It is important to understand how cotangent behaves, especially in different quadrants, as this affects the values returned by its inverse function. The cotangent function is periodic and has specific asymptotes where it is undefined.

Radian Mode in Calculators

Calculators can operate in different modes, with radian mode being crucial for trigonometric calculations in higher mathematics. When in radian mode, angles are measured in radians rather than degrees, which is essential for accurately computing values of trigonometric functions and their inverses. Understanding how to switch between modes and the implications of each is vital for correct calculations.