# Is it possible to have a cosine of 1.0000

I was given a triangle:

side opposite of angle A: unknown, referred to as L

side adjacent of angle A: 11'

hypotenuse: 14'

I have to find the cosine of angle A, the degrees, and the length of "L", opposite angle A, in feet.

L is equal to 8.6602, which in feet is 8'-7 7/16. I did the math, and got angle A= 38 degrees

(my instructor wants length in feet-inches-fraction, I'm in pipefitters school. He also doesn't want minutes or seconds for the angles)

However, when I did the cosine function, I ended up with .999905 We are supposed to round decimals to the fourth place. However, that leaves me with

.9999

which should round up to

1

But...what I'm wondering is can I even have a whole number for a cosine, or should I leave it at .9999?

If .9999 is the mathematically correct answer, I need to know why, in order to defend it in class. If 1 is the correct answer, then I'd just like to know why. However, I asked Google and haven't been able to get an answer that I understand.

• The cosine of 0 degrees is 1. To see this, draw a right angle "triangle" of side lengths 0, 1, 1... – Zhen Lin Mar 10 '12 at 20:59
• The cosine function of what? There is nothing in the picture that has cosine equal to $0.999905$. The cosine of an angle in a triangle cannot be $1$. Cosines near $1$ are for angles fairly close to $0$. – André Nicolas Mar 10 '12 at 21:03
• If you have a right angled triangle with: angle A, side length opposite A is L, side length adjacent to A is 11 and hypotenuse is 14, then I would say that $\cos(A) = 11 / 14= 0.7857...$. And so the angle $A$ is $\cos^{-1}(0.7857) = 48.17$ degrees. I don't understand where your 0.9999 comes from... – Thomas Mar 10 '12 at 21:23
• Oh, and if you round $0.999905$ to the forth place then you get $0.9999$ which isn't equal to $1$. – Thomas Mar 10 '12 at 21:27
• Well your L is correct and $\displaystyle \arccos\left(\frac{11}{14}\right)\approx 0.789774$ is indeed approximatively $38.21$ degrees but it seems to me that the $\cos$ should be $\frac{11}{14}=\cos(0.789774...)$ no? Where did you evaluate the $\cos$ function? – Raymond Manzoni Mar 10 '12 at 21:29

You found the cosine of $11/14$ degrees. But you needed the arccosine (not the cosine) of $11/14$. Just $11/14$, not $11/14$ degrees. The calculator in front of me gives $$\cos^{-1} \frac{11}{14} \approx 38.2^\circ.$$

The cosine of $11/14$ degrees is about $0.9999059$, but that's not what you need here.

• And $11/14$ degrees is equal to about 0.0137, which is indeed very close to zero. – ShreevatsaR Mar 11 '12 at 6:00