Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.R.38

Chapter 3, Problem 2.R.38

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. tan θ = 1.3763819

Verified step by step guidance

Recall that the tangent function, \(\tan \theta\), is periodic with a period of \(180^\circ\). This means if \(\theta\) is a solution, then \(\theta + 180^\circ\) is also a solution within the interval \([0^\circ, 360^\circ)\).

To find the principal angle \(\theta\), use the inverse tangent function: \(\theta = \tan^{-1}(1.3763819)\). This will give you the first angle in degrees.

Make sure your calculator is set to degree mode before calculating the inverse tangent to get the angle in degrees.

The second angle that satisfies the equation in the interval \([0^\circ, 360^\circ)\) is found by adding \(180^\circ\) to the first angle: \(\theta_2 = \theta_1 + 180^\circ\).

Round both angles to the nearest degree as required, and verify that both angles lie within the interval \([0^\circ, 360^\circ)\).

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Key Concepts

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

Tangent Function and Its Properties

The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. It is periodic with a period of 180°, meaning tan(θ) = tan(θ + 180°). Understanding this periodicity helps find multiple angle solutions within a given interval.

Inverse Tangent (Arctan) and Angle Calculation

The inverse tangent function, arctan, is used to find the angle whose tangent value is given. Since arctan returns values typically between -90° and 90°, additional steps are needed to find all solutions in the specified interval, especially considering the tangent function's periodicity.

Interval Restriction and Multiple Solutions

When solving trigonometric equations within a specific interval, such as [0°, 360°), it is important to identify all angles that satisfy the equation within that range. For tangent, two solutions exist per 360° interval, separated by 180°, which must be accounted for when listing all valid angles.

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