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. Recall that the cosecant function is the reciprocal of the sine function. So, \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
2. Identify the angle given: \( \theta = -9.4946 \) radians. Since the angle is in radians, ensure your calculator is set to radian mode.
3. Calculate \( \sin(-9.4946) \) using your calculator or a computational tool.
4. Find the reciprocal of the sine value calculated in the previous step to get \( \csc(-9.4946) = \frac{1}{\sin(-9.4946)} \).
5. Round the result to four decimal places to obtain the final approximation.

Key Concepts

Circular Functions and Their Definitions

Circular functions, such as sine, cosine, and cosecant, relate angles to ratios of sides in a right triangle or coordinates on the unit circle. Specifically, cosecant (csc) is the reciprocal of sine, defined as csc(θ) = 1/sin(θ). Understanding this relationship is essential for evaluating csc at any angle.

Evaluating Trigonometric Functions at Negative Angles

Trigonometric functions have specific properties when dealing with negative angles. For sine, sin(−θ) = −sin(θ), which affects the value of cosecant since it depends on sine. Recognizing how negative angles influence function values helps in correctly computing csc(−9.4946).

Using Calculators for Approximation and Radian Mode

Calculators can approximate trigonometric values to desired decimal places, but the angle mode (degrees or radians) must be set correctly. Since the angle −9.4946 is likely in radians, the calculator should be in radian mode to get an accurate approximation of csc(−9.4946).