Question
Which of the following trigonometric functions, restricted to 0 less than x less than pi, have values greater than 1? Select all that apply.
(1 point)
Responses
f left parenthesis x right parenthesis equals sine x
Image with alt text: f left parenthesis x right parenthesis equals sine x
f left parenthesis x right parenthesis equals cosine x
Image with alt text: f left parenthesis x right parenthesis equals cosine x
f left parenthesis x right parenthesis equals cotangent x
Image with alt text: f left parenthesis x right parenthesis equals cotangent x
f left parenthesis x right parenthesis equals secant x
Image with alt text: f left parenthesis x right parenthesis equals secant x
(1 point)
Responses
f left parenthesis x right parenthesis equals sine x
Image with alt text: f left parenthesis x right parenthesis equals sine x
f left parenthesis x right parenthesis equals cosine x
Image with alt text: f left parenthesis x right parenthesis equals cosine x
f left parenthesis x right parenthesis equals cotangent x
Image with alt text: f left parenthesis x right parenthesis equals cotangent x
f left parenthesis x right parenthesis equals secant x
Image with alt text: f left parenthesis x right parenthesis equals secant x
Answers
Answered by
GPT-4o mini
To determine which of the given trigonometric functions have values greater than 1 when restricted to the interval \(0 < x < \pi\), we can analyze each function.
1. **Sine Function: \(f(x) = \sin(x)\)**
The sine function oscillates between 0 and 1 within the interval \(0 < x < \pi\). Therefore, \(\sin(x)\) is never greater than 1 in this interval.
2. **Cosine Function: \(f(x) = \cos(x)\)**
The cosine function also oscillates between -1 and 1. Within the interval \(0 < x < \pi\), \(\cos(x)\) decreases from 1 (at \(x = 0\)) to -1 (at \(x = \pi\)). Thus, \(\cos(x)\) is also never greater than 1 in this interval.
3. **Cotangent Function: \(f(x) = \cot(x)\)**
The cotangent function is defined as \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). In the interval \(0 < x < \pi\), \(\cot(x)\) changes from positive infinity (as \(x\) approaches 0 from the right) to negative infinity (as \(x\) approaches \(\pi\) from the left). This means \(\cot(x)\) can be greater than 1 for certain values in this interval.
4. **Secant Function: \(f(x) = \sec(x)\)**
The secant function is defined as \(\sec(x) = \frac{1}{\cos(x)}\). For \(0 < x < \frac{\pi}{2}\), \(\cos(x)\) is positive and decreases from 1 to 0, making \(\sec(x)\) greater than 1. For \(\frac{\pi}{2} < x < \pi\), \(\cos(x)\) is negative, resulting in a negative value for \(\sec(x)\). Hence, secant is greater than 1 for part of the interval.
Based on this analysis:
- **Sine function: Not greater than 1.**
- **Cosine function: Not greater than 1.**
- **Cotangent function: May be greater than 1.**
- **Secant function: Greater than 1 in a part of the interval.**
Thus, the correct selections are:
- Cotangent \(f(x) = \cot(x)\)
- Secant \(f(x) = \sec(x)\) (in part of the interval)
### Final Answer:
- **Cotangent \(f(x) = \cot(x)\)**
- **Secant \(f(x) = \sec(x)\)**
1. **Sine Function: \(f(x) = \sin(x)\)**
The sine function oscillates between 0 and 1 within the interval \(0 < x < \pi\). Therefore, \(\sin(x)\) is never greater than 1 in this interval.
2. **Cosine Function: \(f(x) = \cos(x)\)**
The cosine function also oscillates between -1 and 1. Within the interval \(0 < x < \pi\), \(\cos(x)\) decreases from 1 (at \(x = 0\)) to -1 (at \(x = \pi\)). Thus, \(\cos(x)\) is also never greater than 1 in this interval.
3. **Cotangent Function: \(f(x) = \cot(x)\)**
The cotangent function is defined as \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). In the interval \(0 < x < \pi\), \(\cot(x)\) changes from positive infinity (as \(x\) approaches 0 from the right) to negative infinity (as \(x\) approaches \(\pi\) from the left). This means \(\cot(x)\) can be greater than 1 for certain values in this interval.
4. **Secant Function: \(f(x) = \sec(x)\)**
The secant function is defined as \(\sec(x) = \frac{1}{\cos(x)}\). For \(0 < x < \frac{\pi}{2}\), \(\cos(x)\) is positive and decreases from 1 to 0, making \(\sec(x)\) greater than 1. For \(\frac{\pi}{2} < x < \pi\), \(\cos(x)\) is negative, resulting in a negative value for \(\sec(x)\). Hence, secant is greater than 1 for part of the interval.
Based on this analysis:
- **Sine function: Not greater than 1.**
- **Cosine function: Not greater than 1.**
- **Cotangent function: May be greater than 1.**
- **Secant function: Greater than 1 in a part of the interval.**
Thus, the correct selections are:
- Cotangent \(f(x) = \cot(x)\)
- Secant \(f(x) = \sec(x)\) (in part of the interval)
### Final Answer:
- **Cotangent \(f(x) = \cot(x)\)**
- **Secant \(f(x) = \sec(x)\)**