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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 2.3.38
Chapter 3, Problem 2.3.38

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.
cos θ = 0.85536428

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1
Identify the given equation: \(\cos \theta = 0.85536428\) and the interval for \(\theta\) is \([0^\circ, 90^\circ)\).
Recall that the cosine function is positive and decreasing in the first quadrant, so there will be one solution for \(\theta\) in the given interval.
Use the inverse cosine function to find \(\theta\): \(\theta = \cos^{-1}(0.85536428)\).
Calculate the value of \(\theta\) using a calculator set to degree mode to get the angle in degrees.
Round the result to six decimal places as required.

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Key Concepts

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

Cosine Function and Its Properties

The cosine function relates an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. It is periodic and ranges between -1 and 1. Understanding its behavior on the interval [0°, 90°) is essential for finding the angle corresponding to a given cosine value.
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Graph of Sine and Cosine Function

Inverse Cosine (Arccos) Function

The inverse cosine function, denoted arccos or cos⁻¹, returns the angle whose cosine is a given number. It is used to find the angle θ when cos θ is known, typically producing results in the range [0°, 180°]. For this problem, restricting the solution to [0°, 90°) is important.
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Inverse Cosine

Decimal Degree Precision

Decimal degree precision refers to expressing angles in degrees with decimal points, allowing for more exact measurements. Here, answers must be given to six decimal places, which requires careful calculation and rounding to ensure accuracy.
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Converting between Degrees & Radians
Related Practice
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)

Textbook Question

(Modeling) Grade Resistance Solve each problem. See Example 3. A car traveling on a -3° downhill grade has a grade resistance of -145 lb. Determine the weight of the car to the nearest hundred pounds.

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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. (-4, 0)

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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.

csc θ = 1.3861147

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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.

sin θ = 0.84802194

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Textbook Question

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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