Find a value of θ, in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places.csc θ = 9.5670466

Verified step by step guidance

1. Recall that the cosecant function is the reciprocal of the sine function, so \( \csc \theta = \frac{1}{\sin \theta} \).
2. Set up the equation \( \csc \theta = 9.5670466 \) which implies \( \frac{1}{\sin \theta} = 9.5670466 \).
3. Solve for \( \sin \theta \) by taking the reciprocal of both sides: \( \sin \theta = \frac{1}{9.5670466} \).
4. Use a calculator to find the value of \( \sin \theta \).
5. Finally, use the inverse sine function (\( \sin^{-1} \)) to find \( \theta \) in degrees, ensuring that \( \theta \) is within the interval [0°, 90°).

Key Concepts

Cosecant Function

The cosecant function, denoted as csc(θ), is the reciprocal of the sine function. It is defined as csc(θ) = 1/sin(θ). This means that to find the angle θ when given csc(θ), one must first determine sin(θ) by taking the reciprocal of the given value. Understanding this relationship is crucial for solving problems involving cosecant.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, are used to find angles when the value of a trigonometric function is known. For example, if sin(θ) is known, θ can be found using θ = arcsin(value). In this case, after calculating sin(θ) from csc(θ), the angle can be determined using the inverse sine function, ensuring the angle lies within the specified interval.

Angle Measurement in Degrees

In trigonometry, angles can be measured in degrees or radians. The problem specifies that the answer should be in decimal degrees, which means understanding how to convert between radians and degrees may be necessary. The interval [0°, 90°) indicates that the solution must be a positive acute angle, reinforcing the need to ensure the calculated angle falls within this range.