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Sagot :
Let's analyze each statement given by the students about the trigonometric ratios cosecant, secant, and cotangent for an acute angle (an angle between 0° and 90°).
1. Anik's statement:
- "The cosecant, secant, and cotangent of an acute angle may be greater than 1 or less than 1."
- For an acute angle:
- Cosecant (csc): [tex]\( csc(\theta) = \frac{1}{\sin(\theta)} \)[/tex]
- Since [tex]\(\sin(\theta)\)[/tex] for an acute angle is between 0 and 1, [tex]\(\frac{1}{\sin(\theta)}\)[/tex] will always be greater than 1. So Anik is incorrect about cosecant being less than 1.
- Secant (sec): [tex]\( sec(\theta) = \frac{1}{\cos(\theta)} \)[/tex]
- Since [tex]\(\cos(\theta)\)[/tex] for an acute angle is between 0 and 1, [tex]\(\frac{1}{\cos(\theta)}\)[/tex] will always be greater than 1. So Anik is incorrect about secant being less than 1.
- Cotangent (cot): [tex]\( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)[/tex]
- For acute angles, cotangent can be less than 1 (for angles greater than 45°), equal to 1 (at 45°), or greater than 1 (for angles less than 45°). Anik is correct about cotangent.
2. Isabella's statement:
- "The cosecant, secant, and cotangent of an acute angle are always greater than 1."
- Based on our analysis:
- Cosecant and Secant are always greater than 1 for an acute angle.
- Cotangent is not always greater than 1 (since it depends on whether the angle is less than, equal to, or greater than 45°). Isabella is incorrect about cotangent.
3. Morris's statement:
- "The cosecant and secant of an acute angle are always greater than 1, but the cotangent can be greater than 1, less than 1, or equal to 1."
- This aligns perfectly with our analysis:
- Cosecant and Secant are indeed always greater than 1.
- Cotangent can be greater than 1 (for angles less than 45°), equal to 1 (for 45°), or less than 1 (for angles greater than 45°). Morris is correct.
4. Julie's statement:
- "The cosecant and secant of an acute angle may be greater than or equal to 1, but the cotangent of an acute angle is always less than 1."
- While this statement about cotangent being always less than 1 is incorrect (cotangent is less than 1 for angles greater than 45°, but greater than 1 for angles less than 45°, and equal to 1 for 45°).
- Cosecant and Secant are always greater than 1, not just greater than or equal to 1.
From our analysis, the correct student is:
Morris is correct.
1. Anik's statement:
- "The cosecant, secant, and cotangent of an acute angle may be greater than 1 or less than 1."
- For an acute angle:
- Cosecant (csc): [tex]\( csc(\theta) = \frac{1}{\sin(\theta)} \)[/tex]
- Since [tex]\(\sin(\theta)\)[/tex] for an acute angle is between 0 and 1, [tex]\(\frac{1}{\sin(\theta)}\)[/tex] will always be greater than 1. So Anik is incorrect about cosecant being less than 1.
- Secant (sec): [tex]\( sec(\theta) = \frac{1}{\cos(\theta)} \)[/tex]
- Since [tex]\(\cos(\theta)\)[/tex] for an acute angle is between 0 and 1, [tex]\(\frac{1}{\cos(\theta)}\)[/tex] will always be greater than 1. So Anik is incorrect about secant being less than 1.
- Cotangent (cot): [tex]\( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)[/tex]
- For acute angles, cotangent can be less than 1 (for angles greater than 45°), equal to 1 (at 45°), or greater than 1 (for angles less than 45°). Anik is correct about cotangent.
2. Isabella's statement:
- "The cosecant, secant, and cotangent of an acute angle are always greater than 1."
- Based on our analysis:
- Cosecant and Secant are always greater than 1 for an acute angle.
- Cotangent is not always greater than 1 (since it depends on whether the angle is less than, equal to, or greater than 45°). Isabella is incorrect about cotangent.
3. Morris's statement:
- "The cosecant and secant of an acute angle are always greater than 1, but the cotangent can be greater than 1, less than 1, or equal to 1."
- This aligns perfectly with our analysis:
- Cosecant and Secant are indeed always greater than 1.
- Cotangent can be greater than 1 (for angles less than 45°), equal to 1 (for 45°), or less than 1 (for angles greater than 45°). Morris is correct.
4. Julie's statement:
- "The cosecant and secant of an acute angle may be greater than or equal to 1, but the cotangent of an acute angle is always less than 1."
- While this statement about cotangent being always less than 1 is incorrect (cotangent is less than 1 for angles greater than 45°, but greater than 1 for angles less than 45°, and equal to 1 for 45°).
- Cosecant and Secant are always greater than 1, not just greater than or equal to 1.
From our analysis, the correct student is:
Morris is correct.
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