Question

unit: trigonometry progress: the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. angle c is an acute angle with \\( \sec c = \frac{9}{4} \\) and \\( \sin c = \frac{\sqrt{65}}{9} \\). what are the values of \\( \cos c \\) and \\( \csc c \\)? \\( \cos c = \frac{\sqrt{65}}{4}, \csc c = \frac{9}{\sqrt{65}} \\) \\( \cos c = \frac{4}{9}, \csc c = \frac{9}{\sqrt{65}} \\) \\( \cos c = \frac{4}{9}, \csc c = \frac{4}{\sqrt{65}} \\) \\( \cos c = \frac{9}{\sqrt{65}}, \csc c = \frac{4}{9} \\)

Explanation:

Step1: Recall reciprocal identities

Recall that \(\sec\theta=\frac{1}{\cos\theta}\) and \(\csc\theta = \frac{1}{\sin\theta}\).

Step2: Find \(\cos C\) using \(\sec C\)

Given \(\sec C=\frac{9}{4}\), since \(\sec C=\frac{1}{\cos C}\), then \(\cos C=\frac{1}{\sec C}=\frac{4}{9}\).

Step3: Find \(\csc C\) using \(\sin C\)

Given \(\sin C = \frac{\sqrt{65}}{9}\), since \(\csc C=\frac{1}{\sin C}\), then \(\csc C=\frac{1}{\frac{\sqrt{65}}{9}}=\frac{9}{\sqrt{65}}\) (we can also rationalize the denominator, but let's check the options. Wait, wait, no, wait the given \(\sin C\) in the problem? Wait, the problem says "Angle C is an acute angle with \(\sec C=\frac{9}{4}\) and \(\sin C=\frac{\sqrt{65}}{9}\)". Wait, no, maybe I misread. Wait, let's re - check. Wait, the options: let's see the first option is \(\cos C=\frac{\sqrt{65}}{4},\csc C=\frac{9}{\sqrt{65}}\) – no. Wait, wait, maybe I made a mistake. Wait, \(\sec C=\frac{9}{4}=\frac{hypotenuse}{adjacent}\) in a right - triangle. Then, by Pythagoras, opposite side \(=\sqrt{9^{2}-4^{2}}=\sqrt{81 - 16}=\sqrt{65}\). So \(\sin C=\frac{opposite}{hypotenuse}=\frac{\sqrt{65}}{9}\), and \(\cos C=\frac{adjacent}{hypotenuse}=\frac{4}{9}\), and \(\csc C=\frac{1}{\sin C}=\frac{9}{\sqrt{65}}\). Wait, but looking at the options, the third option (let's count the options: first option: \(\cos C=\frac{\sqrt{65}}{4},\csc C=\frac{9}{\sqrt{65}}\); second option: \(\cos C=\frac{4}{9},\csc C=\frac{9}{\sqrt{65}}\); third option: \(\cos C=\frac{4}{9},\csc C=\frac{9}{\sqrt{65}}\) – wait, the second option in the list (the third circle? Wait, the options are:

1. \(\cos C=\frac{\sqrt{65}}{4},\csc C=\frac{9}{\sqrt{65}}\)

1. \(\cos C=\frac{4}{9},\csc C=\frac{9}{\sqrt{65}}\) (wait, the second option as per the image: the second option (the second circle) is \(\cos C=\frac{4}{9},\csc C=\frac{9}{\sqrt{65}}\)? Wait, no, the user's image: let's re - parse. The options are:

• First option: \(\cos C=\frac{\sqrt{65}}{4},\csc C=\frac{9}{\sqrt{65}}\)

• Second option: \(\cos C=\frac{4}{9},\csc C=\frac{4}{\sqrt{65}}\)

• Third option: \(\cos C=\frac{4}{9},\csc C=\frac{9}{\sqrt{65}}\)

• Fourth option: \(\cos C=\frac{9}{\sqrt{65}},\csc C=\frac{4}{9}\)

Wait, my calculation: \(\cos C=\frac{4}{9}\) (from \(\sec C = 9/4\)) and \(\csc C=\frac{9}{\sqrt{65}}\) (from \(\sin C=\sqrt{65}/9\)). So the third option (the one with \(\cos C = 4/9\) and \(\csc C=9/\sqrt{65}\)) is correct. Wait, but in the step - by - step, let's correct:

Wait, \(\sec C=\frac{9}{4}\), so \(\cos C=\frac{4}{9}\) (since \(\sec\theta = 1/\cos\theta\)). \(\sin C=\frac{\sqrt{65}}{9}\), so \(\csc C=\frac{1}{\sin C}=\frac{9}{\sqrt{65}}\). So the option where \(\cos C=\frac{4}{9}\) and \(\csc C=\frac{9}{\sqrt{65}}\) is the correct one. Looking at the options, the third option (the third circle) has \(\cos C=\frac{4}{9},\csc C=\frac{9}{\sqrt{65}}\) (wait, the third option in the list: let's count the options as per the image:

1. \(\cos C=\frac{\sqrt{65}}{4},\csc C=\frac{9}{\sqrt{65}}\)

1. \(\cos C=\frac{4}{9},\csc C=\frac{4}{\sqrt{65}}\)

1. \(\cos C=\frac{4}{9},\csc C=\frac{9}{\sqrt{65}}\)

1. \(\cos C=\frac{9}{\sqrt{65}},\csc C=\frac{4}{9}\)

So the third option is correct.

Answer:

The option with \(\cos C=\frac{4}{9},\csc C=\frac{9}{\sqrt{65}}\) (the third option in the list of options provided)