A guy wire is attached to the top of a 75 m tower and meets the ground at a 65 degree angle. How long is the wire?

For this problem, you would use trigonometry. 

First draw a diagram for the problem; you should have a right triangle with the guy wire as the hypotenuse, 75m as the height (opposite side), and 65 degrees as the angle of the guy wire to the ground (hypotenuse to adjacent side). 

Now use the fact that the sine of an angle in a right triangle is its opposite side divided by its hypotenuse. 

So for this problem: 

Let x = the length of the guy wire. 

sin(65) = 75/ x 

sin(65) * x = 75

75/sin(65) = x

x = $${\frac{{\mathtt{75}}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{65}}^\circ\right)}}} = {\mathtt{82.753\: \!343\: \!922\: \!154\: \!920\: \!5}}$$

x = 82.753 m 

*LOL CPhill strikes again! 

The wire forms a hypotenuse of a right triangle....so we have

sin65 = 75 / x     where x is the wire length, therefore

x = 75/sin65 = about 82.75 m