Question

arun: the cosine and, secant, and cotangent of an acute angle may be greater than 1 or less than 1.
isabella: the cosine, secant, and cotangent of an acute angle are always greater than 1.
kayla: the cosine and sine of an acute angle are always greater than 1, but the cotangent can be greater than 1, less than 1 or equal to 1.
maria: the cosine and sine of an acute angle may be greater than or equal to 1, but the cotangent of an acute angle is always less than 1.
which student is correct?
(1) arun
(2) isabella
(3) kayla
(4) maria

Explanation:

To determine the correct statement, recall the definitions of trigonometric ratios (sine, cosine, tangent) for acute angles (between \(0^\circ\) and \(90^\circ\)):

• Sine (\(\sin\)) and Cosine (\(\cos\)): For an acute angle \(\theta\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Since the hypotenuse is the longest side in a right triangle, \(\sin\theta\) and \(\cos\theta\) are always less than \(1\) (because the numerator is shorter than the hypotenuse).
• Tangent (\(\tan\)): \(\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{\text{opposite}}{\text{adjacent}}\). For acute angles, the opposite side can be shorter than, equal to, or longer than the adjacent side (e.g., \(\tan 45^\circ = 1\), \(\tan 30^\circ=\frac{1}{\sqrt{3}}<1\), \(\tan 60^\circ=\sqrt{3}>1\)). Thus, \(\tan\theta\) can be greater than \(1\), less than \(1\), or equal to \(1\).

Now analyze the options:

• Arun: Claims sine, secant, cotangent can be \(>1\) or \(<1\). But \(\sin\theta<1\) (always), so Arun is incorrect.
• Induha: Claims sine, secant, cotangent are always \(>1\). But \(\sin\theta<1\) (always), so Induha is incorrect.
• Riya: Claims sine and secant are always \(>1\) (false, \(\sin\theta<1\)) and cotangent can vary (but secant is \(\frac{1}{\cos\theta}\), and \(\cos\theta<1\) implies \(\sec\theta>1\) always). So Riya’s first claim is wrong.
• Maria: States sine and cosine of an acute angle are always \(<1\) (correct, as hypotenuse is longest), and tangent can be \(>1\), \(<1\), or \(=1\) (correct, as shown). This matches the trigonometric definitions.

Answer:

Maria (assuming the last option is labeled as Maria, e.g., if the options are A. Arun, B. Induha, C. Riya, D. Maria, then the answer is D. Maria)