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Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblem 2.3.30
Chapter 3, Problem 2.3.30

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

Verified step by step guidance
1
Understand that the problem asks for an angle \( \theta \) in the interval \( [0^\circ, 90^\circ) \) such that \( \tan \theta = 6.4358841 \). The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side.
Recall that to find \( \theta \) when given \( \tan \theta \), you use the inverse tangent function (also called arctangent), denoted as \( \tan^{-1} \) or \( \arctan \). This function returns the angle whose tangent is the given value.
Set up the equation: \[ \theta = \tan^{-1}(6.4358841) \]. This will give the angle in degrees if your calculator or software is set to degree mode.
Use a calculator or computational tool to evaluate \( \tan^{-1}(6.4358841) \) and obtain the angle in decimal degrees. Make sure the calculator is in degree mode to get the correct unit.
Since the problem asks for the answer to six decimal places, round the result from the inverse tangent calculation to six decimal places. This will be your value of \( \theta \) in the interval \( [0^\circ, 90^\circ) \).

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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 of an angle in a right triangle is the ratio of the opposite side to the adjacent side. It is also defined as tan θ = sin θ / cos θ. Understanding the tangent function is essential for solving equations involving tan θ, such as finding the angle when the tangent value is given.
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Inverse Tangent (Arctan) Function

The inverse tangent function, denoted arctan or tan⁻¹, is used to find the angle whose tangent is a given number. It returns an angle typically in the range (-90°, 90°). Using arctan allows us to determine θ when tan θ is known, which is crucial for solving the given problem.
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Angle Measurement and Interval Restrictions

Angles can be measured in degrees or radians, and problems often specify intervals for valid solutions. Here, θ must lie in [0°, 90°), meaning the angle is in the first quadrant where tangent values are positive. Recognizing this interval helps select the correct angle from the inverse tangent output.
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