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Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblema 2.3.68
Capitolo 3, Problema 2.3.68

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. tan θ = 0.70020753

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1
Recall that the tangent function, \(\tan \theta\), is periodic with period \(180^\circ\), meaning that 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: \(\theta = \tan^{-1}(0.70020753)\). 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.
Once you have the first angle \(\theta_1\), find the second angle by adding \(180^\circ\) to it: \(\theta_2 = \theta_1 + 180^\circ\).
Verify that both \(\theta_1\) and \(\theta_2\) lie within the interval \([0^\circ, 360^\circ)\). These two angles are the solutions to the equation \(\tan \theta = 0.70020753\).

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Understanding the Tangent Function

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. On the unit circle, tan θ is the ratio of the y-coordinate to the x-coordinate. It is periodic with a period of 180°, meaning tan(θ) = tan(θ + 180°).
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Introduction to Tangent Graph

Solving Trigonometric Equations in a Given Interval

To find all solutions for tan θ = k within [0°, 360°), identify the principal angle using the inverse tangent function, then use the periodicity of tangent to find the second solution by adding 180°. Both solutions must lie within the specified interval.
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How to Solve Linear Trigonometric Equations

Using Inverse Trigonometric Functions and Rounding

The inverse tangent function (arctan or tan⁻¹) returns the principal angle whose tangent is the given value, typically in (-90°, 90°). After calculating, round the angle to the nearest degree as required, ensuring the final answers fit the problem's interval.
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