Given that start fraction pi over 2 end fraction less than or equal to x less than or equal to pi, which of the following are positive? Select all that apply.

Question

(1 point)
Responses

cos x
cos x

sin x
sin x

tan x
tan x

sec x
sec x

csc x
csc x

cot x
cot x
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Answer 1

Given the interval \(\frac{\pi}{2} \leq x \leq \pi\), we can analyze the signs of the trigonometric functions:

1. \(\cos x\): The cosine function is negative in the second quadrant. Therefore, \(\cos x < 0\).

2. \(\sin x\): The sine function is positive in the second quadrant. Therefore, \(\sin x > 0\).

3. \(\tan x\): The tangent function is defined as \(\tan x = \frac{\sin x}{\cos x}\). Since \(\sin x\) is positive and \(\cos x\) is negative, \(\tan x < 0\).

4. \(\sec x\): The secant function is the reciprocal of cosine, \(\sec x = \frac{1}{\cos x}\). Since \(\cos x < 0\), \(\sec x < 0\).

5. \(\csc x\): The cosecant function is the reciprocal of sine, \(\csc x = \frac{1}{\sin x}\). Since \(\sin x > 0\), \(\csc x > 0\).

6. \(\cot x\): The cotangent function is defined as \(\cot x = \frac{\cos x}{\sin x}\). Since \(\cos x < 0\) and \(\sin x > 0\), \(\cot x < 0\).

In summary:

• \(\cos x < 0\)
• \(\sin x > 0\)
• \(\tan x < 0\)
• \(\sec x < 0\)
• \(\csc x > 0\)
• \(\cot x < 0\)

Thus, the functions that are positive in the interval \(\frac{\pi}{2} \leq x \leq \pi\) are:

• \(\sin x\)
• \(\csc x\)

So the correct responses are:

• \(\sin x\)
• \(\csc x\)