Which trigonometric function has a range that does not include 0.4?

Question

(1 point)
Responses

y = cos x
y = cos x

y = cot x
y = cot x

y = csc x
y = csc x

y = sin x
y = sin x

Answer 1

To determine which trigonometric function has a range that does not include 0.4, let's analyze the ranges of the given functions:

1. y = cos x: The range of the cosine function is from -1 to 1, so it does include 0.4.

2. y = cot x: The cotangent function has a range of all real numbers (from -∞ to +∞), so it certainly includes 0.4.

3. y = csc x: The cosecant function is defined as the reciprocal of the sine function. Since the sine function has a range from -1 to 1, the cosecant function's range is (-∞, -1] ∪ [1, +∞). Therefore, it does not include 0.4.

4. y = sin x: The range of the sine function is from -1 to 1, so it does include 0.4.

From this analysis, the only function whose range does not include 0.4 is:

y = csc x.