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Four students wrote statements about cosecant, secant, and cotangent values as shown below.

\begin{tabular}{|l|l|}
\hline
Anik & The cosecant, secant, and cotangent of an acute angle may be greater than 1 or less than 1. \\
\hline
Isabella & The cosecant, secant, and cotangent of an acute angle are always greater than 1. \\
\hline
Morris & 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. \\
\hline
Sam & 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. \\
\hline
\end{tabular}

Which student is correct?
A. Anik
B. Isabella
C. Morris
D. Sam

Sagot :

GinnyAnswer
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.
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