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 2.3.68

Chapter 3, Problem 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

Verified step by step guidance

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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Key Concepts

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

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°).

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.

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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Textbook Question

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (2, 2)

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Textbook Question

Solve each problem. See Examples 1 and 2. Distance Traveled by a Ship A ship travels 55 km on a bearing of 27° and then travels on a bearing of 117° for 140 km. Find the distance from the starting point to the ending point.

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Textbook Question

Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.

cos(90°-3.69°)

Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.

cos 41° 24'

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Textbook Question

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.

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