How to... find the size of an interior angle of a right triangle using sin/cos/tan

So yesterday's post was working with right triangles and trig functions when you know the interior angles and you have the length of one side of the triangle, but what about if you want to find the size of an interior angle? Well, we use the same trig functions - sin/cos/tan - as we do if we're trying to find the length of a side except we use the inverse.


By using the inverse functions we can find the size of the interior angle. It's important when you're doing this to make sure you still get your SOH CAH TOA correct and pay special attention to which side is the opposite (O) or adjacent (A) side in relation to whichever angle you're wanting to find. It's a common mistake to go "oh, I use sine" and then switch up the O, A or H. This is particularly common if there is a drawing and the right triangle is positioned with the hypotenuse facing the bottom of the page, like this...



Anyway, here's an example of using inverse trig functions -

Now, if you put this into your calculator it will probably return the answer 36.8698 etc.

Generally we don't talk about degrees with decimals, but rather as degrees and minutes. Just like with time, there are sixty minutes in one degree as there are sixty minutes in one hour. To change it on your calculator to read as degrees and minutes rather than with a decimal, find the button on your calculator that looks like it is a couple of apostrophes and degrees signs (the little bubble apostrophe thing :) ). It may be different on different calculators, so play around with it and you'll know when you've found it because it will change your answer to read as degrees and minutes. Don't wait for an exam situation to do this :)

Xx

How to... find information about a right triangle using sin/cos/tan

Before you use these rules, it is important to ensure the triangle you are working with is, in fact, a right triangle. Never assume a triangle is a right triangle! This is a rookie mistake. Unless the angle is marked as 90 degrees, assume it is not until you can prove it.

Okay, so if you really do have a right triangle, you can find relationships between the interior angles and the lengths of the sides of the triangle using the trigonometric functions sine, cosine and tangent (i.e. sin, cos and tan).

Now, PLEASE don't judge my handwriting. I just bought a Bamboo pad and pen. I love it, but it feels pretty wonky for now... I'm hoping I'll get better at it once I get more practice in so you won't have to read math tutorials that appear to be written by a 5 year old...
Ergh, I know it looks awful! But that's all there is to it. If you say "SOH CAH TOA" out loud a couple of times it will just be etched into your brain forever. I'm not sure why, but it's one of those things everyone seems to remember... That doesn't necessarily mean people remember what it means or how to use it, but the sounds are easy enough to remember.

Here's an example... Again, excuse my awful scrawl. Practice makes perfect :)




You might have noticed an interesting relationship between sine and cosine. Oh god, so nerdy. Interesting? Yeah, I know what you're thinking. But the more of this stuff you do the more you will realise how everything is related and you just start to see relationships all over the place. Before you know it you'll be going "Ngaaaa" and yelling out "PI IS EXACTLY THREE!" to shock people into being quiet and getting their attention. Yes, the Simpsons episode where Lisa discovers the bully antidote. Anyway, back to the interesting relationship! In the example above, sin40' turns out to be the same as cos50'.

In fact, sin(x)=cos(90-x) and cos(x)=sin(90-x).

Anyway, this post could continue for pages and pages as I go on and on about different trig functions. I'll post some more stuff in the coming days, but for now I hope this has been of some use to you. Please send through any questions you may have. I'm happy to answer or clarify anything.