Practical Trigonometry for the Anal(ytical)

Dear trig-haters,

I got a new TV a few weeks ago along with a couch. Since then, I’ve been living the American dream of lounging about, eating chips, and frying my retina from hours of Redbox movies.

When I initially set up the TV and couch, I measured out the room but just guess-timated how to angle the TV and the couch. Well, one side of the couch became noticeably more worn than the other side (due to my fat-ass), so during my week of laziness, I did one productive thing: measure out exactly where I should place the TV to have it square to the middle of the couch. Now it is perfect as it will be, and I know the exact angle at which the TV should be positioned in relation to the walls: 47.655 degrees.

How did I come up with this number? I’m glad you didn’t ask. I’m going to tell you anyway, but I swear it won’t be that painful.

The TV stand makes a right triangle with the two walls (the stand is in the corner of the living room). The center of the TV stand projects a straight line to the center of the couch, making another right triangle (imaginary line from TV to couch, one wall, the adjacent wall). The cool thing is that the first triangle shares a similar angle as the second triangle.

So I came up with three equations to describe the set-up, simplified them to two equations, graphed it, and found the point of intersection. Voila!

Not that you’re interested, but here’s the proof:

sin(angle) = b/50
sin(angle) = 25/ (b – x)
tan(angle) = 103/ (97 – x)

simplified into two graphable equations (getting rid of b):

x = 50sin(angle) – 25/ sin(angle)
x = (97tan(angle) – 103)/ (tan(angle))

I graphed it using a graphing calculator and found the intersection to be (47.655, 3.131). This means the angle is 47.655.

In more practical terms, the TV stand should be placed so that back corners touch 37 inches on one wall and 33.7 inches on the adjacent wall.

It took me 2 days, a hack method on finding the solution (making a table of values), and recalling knowledge I had 5 years ago in high school to figure out the answer.

Who says trigonometry is a waste of time? I guess those who would just place a TV all willy-nilly in a corner and hope for the best.

Go take an aspirin for the headache I gave you. If you’re interested in the proof, I can scan the diagram which would make more sense.

Learning trigonometry was what I did in high school while all the other guys were out scoring. I think it's worth it now that I'm making 4-5 times their salary.