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 guidance

1. Recognize that \( \cot \theta = \frac{1}{\tan \theta} \). Therefore, \( \tan \theta = \frac{1}{0.21563481} \).
2. Calculate \( \tan \theta \) using the reciprocal of 0.21563481.
3. Use a calculator to find \( \theta \) by taking the arctangent (inverse tangent) of the result from the previous step: \( \theta = \tan^{-1}(\text{result}) \).
4. Ensure that the calculated \( \theta \) is within the interval [0°, 90°).
5. Express the final answer in decimal degrees to six decimal places.

Key Concepts

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function. It is defined as cot(θ) = cos(θ) / sin(θ). In a right triangle, cotangent represents the ratio of the adjacent side to the opposite side. Understanding this function is crucial for solving equations involving cotangent, as it allows us to relate angles to their corresponding trigonometric ratios.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, arcsin, and arccos, are used to find angles when given a trigonometric ratio. For cotangent, the inverse function is arccot or cot^(-1). These functions are essential for determining the angle θ that corresponds to a specific cotangent value, especially when the angle is constrained within a certain interval, like [0°, 90°).

Angle Measurement in Degrees

Angles can be measured in degrees or radians, with degrees being a more common unit in many applications. The interval [0°, 90°) refers to angles from 0 degrees up to, but not including, 90 degrees. When solving trigonometric equations, it is important to express the final answer in the specified unit, ensuring precision, especially when rounding to six decimal places.