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 \(\sin \theta = \frac{1}{\csc \theta}\).
2. Substitute the given value of \(\csc \theta\) into the equation: \(\sin \theta = \frac{1}{1.3861147}\).
3. Calculate the value of \(\sin \theta\) using the reciprocal of \(1.3861147\) (do not compute the final decimal here, just set up the expression).
4. Use the inverse sine function to find \(\theta\): \(\theta = \sin^{-1}(\sin \theta)\), where \(\sin \theta\) is the value found in the previous step.
5. 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.

Key Concepts

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.

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.

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.