Verified answer

sin 15⁰ = 6900 / h

h = 6900 / sin 15⁰

h = 26660 ft is length of trail to nearest foot.

You don't need the law of cosines, just the law of sines.

The trail is the hypotenuse of a right-angled triangle. The angle of the hypotenuse is 15°. The opposite side is 7800 - 900 = 6,900 feet (height difference between lodge and lake). Let the trail length be a, the opposite side be b, and the trail incline be B. A is thus 90°, because it lies opposite the hypotenuse.

a/sin A = b/sin B

a = b sin A/sin B

a = 6,900 × sin 90/sin 15

a = 6,900 × 1/sin 15

a = 26,660 feet.

Ummmmmmmmmm do u need the work proven??? Regulation of sines is a/SinA=b/SinB=c/SinC... (the capital letters are angles and the lowercase ones are aspects go to mode to your calculator and prefer levels... (: i had the same obstacle