Which of the following trigonometric functions has a range of all real numbers? Select all that apply.

(1 point)
Responses

y = sin x
y = sin x

y = cos x
y = cos x

y = tan x
y = tan x

y = cot x
y = cot x

y = csc x
y = csc x

y = sec x

1 answer

The trigonometric functions that have a range of all real numbers are:

  • \( y = \tan x \)
  • \( y = \cot x \)

Both the tangent and cotangent functions can take any real value as \( x \) varies, hence their ranges are all real numbers.

The other functions listed (sine, cosine, cosecant, and secant) have limited ranges:

  • \( y = \sin x \) has a range of \([-1, 1]\).
  • \( y = \cos x \) has a range of \([-1, 1]\).
  • \( y = \csc x \) has a range of \((-\infty, -1] \cup [1, \infty)\).
  • \( y = \sec x \) has a range of \((-\infty, -1] \cup [1, \infty)\).

Therefore, the correct answers are:

  • \( y = \tan x \)
  • \( y = \cot x \)