Question

express \\(\cos i\\) as a fraction in simplest terms.

answer
\\(\cos i = \\)

Explanation:

Step1: Recall cosine definition

In a right triangle, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$. For angle \( I \), the adjacent side is \( HI = 23 \), and the hypotenuse is \( GI = 25 \)? Wait, no, wait. Wait, triangle \( HIG \) is right-angled at \( H \), so sides: \( HI = 23 \), \( HG \) is another leg, \( GI = 25 \) is the hypotenuse? Wait, no, angle \( I \): adjacent side to \( I \) is \( HI \), opposite is \( HG \), hypotenuse is \( GI \). Wait, let's confirm: in right triangle \( H \), right angle at \( H \), so vertices \( H \), \( I \), \( G \). So angle at \( I \): the sides: adjacent is \( HI \) (length 23), hypotenuse is \( GI \) (length 25)? Wait, no, wait, hypotenuse is the side opposite the right angle, so \( GI \) is hypotenuse (length 25), \( HI = 23 \) (leg), \( HG \) (leg). So for angle \( I \), cosine is adjacent over hypotenuse. Adjacent to \( I \) is \( HI = 23 \), hypotenuse is \( GI = 25 \)? Wait, no, wait, maybe I mixed up. Wait, in right triangle, cosine of angle \( \theta \) is (adjacent side) / (hypotenuse). So angle \( I \): the sides: the side adjacent to \( I \) is \( HI \), and the hypotenuse is \( GI \). Wait, but let's check: \( HI = 23 \), \( GI = 25 \). Wait, but is \( HI \) adjacent? Let's visualize: right angle at \( H \), so \( H \) is between \( I \) and \( G \)? No, \( H \), \( I \), \( G \): \( H \) is the right angle, so \( HI \) and \( HG \) are legs, \( GI \) is hypotenuse. So angle at \( I \): the sides: from \( I \), the two sides are \( HI \) (length 23) and \( GI \) (hypotenuse, length 25), and \( HG \) (other leg). So adjacent to angle \( I \) is \( HI \), hypotenuse is \( GI \). Wait, but wait, maybe \( HI \) is not adjacent. Wait, no: angle at \( I \), the sides forming angle \( I \) are \( HI \) and \( GI \), with \( HI \) being one leg, \( GI \) hypotenuse. So adjacent side to \( I \) is \( HI \), hypotenuse is \( GI \). So \( \cos(I) = \frac{HI}{GI} = \frac{23}{25} \)? Wait, but wait, is \( GI \) the hypotenuse? Let's confirm: in right triangle, hypotenuse is the longest side. \( 25 \) is longer than \( 23 \), so yes, \( GI = 25 \) is hypotenuse, \( HI = 23 \) is leg, \( HG \) can be found by Pythagoras: \( HG = \sqrt{25^2 - 23^2} = \sqrt{625 - 529} = \sqrt{96} = 4\sqrt{6} \), but we don't need that. So for angle \( I \), adjacent side is \( HI = 23 \), hypotenuse is \( GI = 25 \). So \( \cos(I) = \frac{23}{25} \).

Step2: Confirm the definition

Yes, in a right triangle, cosine of an acute angle is the ratio of the length of the adjacent side to the length of the hypotenuse. So for angle \( I \), adjacent side is \( HI = 23 \), hypotenuse is \( GI = 25 \), so \( \cos(I) = \frac{23}{25} \).

Answer:

\(\frac{23}{25}\)