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.tan θ = 6.4358841

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Step 1: Understand the problem. We need to find an angle \( \theta \) such that \( \tan \theta = 6.4358841 \) within the interval \([0^\circ, 90^\circ)\).

Step 2: Use the inverse tangent function to find \( \theta \). The inverse tangent function, denoted as \( \tan^{-1} \) or \( \arctan \), will help us find the angle whose tangent is 6.4358841.

Step 3: Calculate \( \theta = \tan^{-1}(6.4358841) \). This will give us the angle in degrees.

Step 4: Ensure the calculated angle \( \theta \) is within the specified interval \([0^\circ, 90^\circ)\). Since the tangent function is positive in the first quadrant, the angle should naturally fall within this range.

Step 5: Round the result to six decimal places to meet the problem's requirements.

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Key Concepts

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

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed as tan(θ) = sin(θ) / cos(θ). Understanding this function is crucial for solving problems involving angles and their corresponding ratios in trigonometry.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan or tan⁻¹, are used to find the angle when the value of a trigonometric function is known. For example, if tan(θ) = 6.4358841, then θ can be found using θ = arctan(6.4358841). This concept is essential for determining angles from given ratios in trigonometric equations.

Angle Measurement in Degrees

Angles can be measured in degrees, with a full circle comprising 360 degrees. In this context, the interval [0°, 90°) indicates that we are looking for angles in the first quadrant, where both sine and cosine values are positive. Understanding how to convert and express angles in decimal degrees is important for precise calculations and answers.

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