How to find an angle (in degrees) in a right triangle, given its sides?

Question

I need to find out a degree of an angle. Pretty simple, or so I thought. I remember doing a crap-ton of these in high-school, sadly the details did not remain.

Anyway, let's take a look at this triangle.

Let's say the angle I want to find out is D. Well, I decide to use sin and assume

sin(D) = 3 / 5
D = asin(0.6) 
D = 64 rad ???

cos(D) = 3/4
D = acos(0.75)
D = 72 rad ???

I feel like I'm missing something very obvious and stupid here. Can't figure out what that is...

Thanks!

Answer 1

There is a confusion here between radians and degrees. $\arcsin(3/5) = 0.643$ radians. But radians are not the same as degrees. You can convert from radians to degrees by multiplying by $\frac{180}{\pi}$.

Here, this gives that $0.643$ radians is the same as about $36.8^\circ$ (and we would get the exact answer if I hadn't truncated to $0.643$ in my calculation).

(And as the others have pointed out, your cosine calculation uses the wrong two legs of the triangle).

Answer 2

The issue here is that $\cos(D)=\dfrac{\text{adjacent}}{\text{hypotenuse}}=\dfrac{4}{5}$ instead of $\cos(D)=\dfrac{3}{4}$. It is the tangent of $D$ which is $\dfrac{3}{4}$.

Note that in your picture you are using degrees, while the calculations have been in radians.

Answer 3

What you're missing is that cosine = adjacent/hypotenuse. Your trig rules:

sine = opposite/hypotenuse

cosine = adjacent/hypotenuse

tangent = sine/cosine = opposite/adjacent

Answer 4

I am not sure if I got your question right but if you want to translate radiants to degrees you can use this formular: $$\frac{\alpha}{360°}=\frac{r}{2\pi}$$

So in your case $r = 0.64$ and thus $\alpha = 36.67°$