Two legs of a right triangle have lengths 14 and 9. The mreasure of the smaller acute angle to the nearest degree is

Question

Two legs of a right triangle have lengths 14 and 9. The measure of the smaller acute angle to the nearest degree is....? Can someone talk me through this? I'm literally stressing out and cannot understand what i'm supposed to do? I do not need the exact answer really, just some help through it, please? I've been all over the internet searching how to figure this out and i'm not understanding. The sin, cos and tan is very confusing to me. Thank you

Answer 1

First remember from geometry that the smallest angle in a triangle is always across from the smallest side.

Also, in a right triangle, the hypotenuse is always the longest side. Therefore we don't need to know the length of the hypotenuse in your triangle since the hypotenuse can't possibly be the shortest side. So the shortest side must be the smaller of the two legs. The smaller of the two legs is the one with length $9$. Therefore the smallest angle in the triangle is the one across from the side of length $9.$

At this point drawing a picture helps. (Picture is not to scale.)

The angle labeled $x$ is the one we want.

Regarding your confusion with sin, cos, tan, the following mnemonic device may help:

$$SOH \quad CAH \quad TOA$$

$SOH$ means "Sine = Opposite over Hypotenuse"

$CAH$ means "Cosine = Aadjacent over Hypotenuse"

$TOA$ means "Tangent = Opposite over Adjacent"

For this problem we'll use the tangent. This is because we want the angle labeled $x$, and we know what the lengths of the sides Opposite $x$ and Adjacent to $x$ are. So we have:

$$\tan x = \frac9{14}$$

This means $x = \tan^{-1} \dfrac9{14}$, where $\tan^{-1}$ represents the inverse tangent function. At this point you'll need a calculator if you want an approximate answer. Make sure your calculator is in degree mode! If it's in radian mode (or any other mode besides degree) then you'll likely get a wrong answer. Anyway, with a calculator we find that $$ x \approx 32.7352^\circ,$$ which, to the nearest degree, is $x \approx 33^\circ$.

Answer 2

Trigonometry is straightforward and easy .... once you learn it.

What we know of right triangles is that your triangle with legs of $9$ and $14$ and hypontenuse, $whatever$ will have some specific angle. And we know that all right triangles with the same angles will have side lengths that are proportional.

And it's pretty clear that as the base angle changes the propotions between sides will change. So there must be some one to one relation between the angles and the proportions of the side lengths.

It seems reasonable to figure out there most be some algebraic way to determine the angle from the proportions and vice versa. There isn't. It can't be done. It can be approximated... but it can't be calculated by algebraic or arithmetic manipulation.

So, it seems reasonable that someone should make a list of all angle values and corresponding proportions. They have. They are call trigonometry tables.

Here's how they work. (Sorry, I haven't really figured out how to do images.)

Imagine you have a right triangle with a base of length $A$; a "height" leg of length $O$; and a hypotenuse of length $h$. Let's imagine the angles are $x$ for the one one the base; 90 (of course); and on the top $90 - x$.

Just to make sure we are on the same page. The angle $x$ is "on the ground", next to or adjacent to the line $A$, and across or opposite the "height" leg of $O$. (Spoiler: I chose the labels $A$ because $A$ stands for "Adjacent" and $O$ because $O$ stands for "Opposite".

We know that the proportion of the adjacent side compare to the hypotenuse will always be the same proportion for any right triange with base angle $x$. If we blew up the size of the triangle $A$ and $h$ would grow proportionally so $A/h$ will always be a certain value based on $x$.

Also we know if $x$ were to get bigger or smaller the proportion would change.

So we know there is a relation between $x \leftarrow\rightarrow A/h$. We call this relation "cosine" and we say $\cos x = A/h$.

All that means is that for an angle $x$, $\cos x$ will be some number, and if you draw the triangle we described, the proportion $A/h$ will be that number $\cos x$.

We define "sine" as $\sin x $ is the proportion $O/h$.

We define "tangent" as $\tan x$ is the proportion $O/A = \frac {O/h}{A/h} = \frac {\sin x}{\cos x}$.

And we rely on the blood and sweat of minimally paid mathematicians of centuries ago who collected all the values and listed them in tables.

....

So in your triangle. $A$ is the side that is $14$ and $O$ is the side that is $9$. And we want to find the angle $x$ where $O/A = 9/14$. In other words, $\tan x = 9/14$. So we go to our tables and look up $9/14$ in the answer and look for the $x$ that gives us that answer.

In other words the inverse.

We can the inverses of cosine, sine and tangent, the arccosine, arcsine and arctangent. So if $\tan x = 9/14$ then $\arctan 9/14 = x$.

And to figure it out we use a calculator. $\arctan 9/14 = 32.735226272107598215662979268428$ so the angle is $\approx 32.7$.

This actually should make a little sense. For an angle $y = 30$ we have that this is a $30-60-90$ so the sides are $O = \frac 12h; A= \frac{\sqrt{3}}2; h= h$. So $O/A = \frac {1/2}{\sqrt{3}/2} = 1/{\sqrt{3}}$ so $\tan 30 = 1/{\sqrt{3}} \approx 0.577$ which is just a tiny bit less than $9/14 \approx 0.643$.