Find a calculator approximation to four decimal places for each circular function value. See Example 3. csc (―9.4946)

Verified step by step guidance

1. Recognize that the cosecant function, \( \csc(\theta) \), is the reciprocal of the sine function, \( \sin(\theta) \). Therefore, \( \csc(-9.4946) = \frac{1}{\sin(-9.4946)} \).
2. Use the property of the sine function that \( \sin(-\theta) = -\sin(\theta) \) to simplify the expression: \( \csc(-9.4946) = \frac{1}{-\sin(9.4946)} \).
3. Calculate \( \sin(9.4946) \) using a calculator to find its approximate value.
4. Take the reciprocal of the calculated sine value to find \( \csc(9.4946) \).
5. Round the result to four decimal places to obtain the final approximation for \( \csc(-9.4946) \).

Key Concepts

Cosecant Function

The cosecant function, denoted as csc, is the reciprocal of the sine function. For any angle θ, csc(θ) = 1/sin(θ). This means that to find the cosecant of an angle, one must first determine the sine of that angle and then take its reciprocal. Understanding this relationship is crucial for calculating csc values accurately.

Calculator Functions

Using a scientific calculator effectively is essential for approximating trigonometric functions. Most calculators have dedicated buttons for sine, cosine, and tangent, as well as their reciprocals. Familiarity with how to input angles in radians or degrees, depending on the calculator's settings, is necessary to obtain correct results for functions like csc.

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius. In trigonometry, many functions are often evaluated in radians, so converting degrees to radians or understanding when to use radians is vital for accurate calculations, especially when dealing with non-standard angles.