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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 2.3.34
Chapter 3, Problem 2.3.34

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

Verified step by step guidance
1
Recall the definition of the cosecant function: \(\csc \theta = \frac{1}{\sin \theta}\). This means that \(\sin \theta = \frac{1}{\csc \theta}\).
Substitute the given value of \(\csc \theta\) into the equation: \(\sin \theta = \frac{1}{1.3861147}\).
Calculate the value of \(\sin \theta\) using the reciprocal of \(1.3861147\) (do not compute the final decimal here, just set up the expression).
Use the inverse sine function to find \(\theta\): \(\theta = \sin^{-1}(\sin \theta)\), where \(\sin \theta\) is the value found in the previous step.
Since the problem restricts \(\theta\) to the interval \([0^\circ, 90^\circ)\), select the principal value of \(\theta\) from the inverse sine calculation and express it in decimal degrees to six decimal places.

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Key Concepts

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

Cosecant Function (csc θ)

The cosecant function is the reciprocal of the sine function, defined as csc θ = 1/sin θ. It is undefined when sin θ = 0 and is used to find angles when the ratio of the hypotenuse to the opposite side is known.
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Graphs of Secant and Cosecant Functions

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find an angle when given a trigonometric ratio. For csc θ, we first find sin θ by taking the reciprocal, then use the inverse sine (arcsin) to determine θ within the specified interval.
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Introduction to Inverse Trig Functions

Angle Measurement and Interval Restrictions

Angles are measured in degrees or radians, and specifying an interval like [0°, 90°) restricts the solution to the first quadrant. This ensures the angle found is within the domain where sine and cosecant are positive.
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Reference Angles on the Unit Circle
Related Practice
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.

cos θ = 0.85536428

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