Question

based on the diagram, pick the two choices below that represent the expression \\(\cos(65)\degree\\). (note: side lengths are rounded for simplicity, so the expressions may only be approximately equal.)
(diagram: a right triangle with one angle 65°, hypotenuse 255, adjacent side to 65° angle is 108)
show your work here
hint: to add trig functions, type sin, cos, tan, ...
options: \\(\frac{38}{85}\\) (maybe typo, original diagram has 108 and 255), \\(\sin(65\degree)\\), \\(\sin(25\degree)\\), \\(\cos(155\degree)\\), \\(\frac{85}{38}\\) (maybe typo)

Explanation:

Step1: Recall cosine in right triangle

In a right triangle, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$. For $\theta = 65^\circ$, adjacent side is 108, hypotenuse is 255. So $\cos(65^\circ)=\frac{108}{255}$.

Step2: Use co - function identity

We know that $\cos(\theta)=\sin(90^\circ - \theta)$. For $\theta = 65^\circ$, $90^\circ-65^\circ = 25^\circ$, so $\cos(65^\circ)=\sin(25^\circ)$. Also, we can check the ratio $\frac{108}{255}$ (simplify or approximate, but the key is the co - function and the adjacent over hypotenuse).

Answer:

The two choices are $\frac{108}{255}$ and $\sin(25^\circ)$ (assuming the options are $\frac{108}{255}$, $\sin(25^\circ)$, $\sin(65^\circ)$, $\cos(155^\circ)$, $\frac{255}{108}$ etc. from the given options in the problem. So the correct options are the one with $\frac{108}{255}$ and $\sin(25^\circ)$).