How To Learn Trigonometry Intuitively

I’m a programmer and I love algebra, but I gave up hope on ever understanding trigonometry beyond simply memorizing percentages. I never realized it was this simple.

Thank you.

Hi Khalid,
Keep helping the world in this Wonderful way.
Thank you very much.
Always waiting for ur new ideas and simplifying the concept.

Hi Khalid,
Keep helping the world in this Wonderful way.
Thank you very much.
Always waiting for ur new ideas in simplifying the concept.


Thanks for the lovely explanation.

Even with the “wall” concept, it’s not intuitive why Tan(x) is a positive function in the 3rd Quadrant. That Tan(x) = Y component / X component or Sine(x)/Cos(x) kinda explains it - that both quantities are negative in the 3rd quadrant and hence the Tan function, which is a ratio, is positive. However, when you visualise the Tan function in the 3rd Quadrant, intuitively it feels like it should be negative.

Added complication is that if you take the word TANGENT literally as a slope of the circle then at 90 degrees the slope of the tangent should be zero. But Tan function is undefined at 90 degrees (division by zero at this point).

While if you visualize Tan as a magnitude/ length of the tangent, then it ought to be negative in the 3rd Quadrant. How can I visualize this better?

@Kai: Always interested in checking out resources, though I’ll have to brush up (i.e. learn) some German!

@krishnamoorthy: Thanks so much. I do think most ideas can be as simple as falling off a log if seen the right way.

@Johan: Thanks! Hah, you should have seen the original post, which was about twice as big :). I’ll be doing a follow-up with some of that content.

@Tom: Thanks. I was in the same boat, thinking I had to memorize everything. It’s almost like refactoring ugly code, sometimes there’s a simpler way to think about an existing problem which makes everything snap together.

@Bhadrasheel: Appreciate it!

@Ellie: Great question – how the trig angles behave in other quadrants is something I’d like to cover in the follow-up. (Article was getting big, something good for the follow-up!)

Using the percentage analogy, tangent is the height relative to the wall distance, but each component can have a sign:

Is the wall in front (positive) or behind (negative)

Is the height above ground (positive) or below ground (negative)

For example,

  • For x = -30, we are pointing “underground” so the tangent is negative.
  • For x = 120, we are pointing “backward”. The height is positive, but we are on the “back wall” so it’s negative.
  • For x = 210, we are pointing underground AND backward. So this is negative height on the back wall, which counts as positive :slight_smile:

This matches the signs for cosine (front wall / back wall) and sine (above ground / underground) so the calculations are the same :).

@kalid: Actually I am doing the same that you are doing, breaking everything down, not taking formula as-is, trying to find the insights behind… but just in German and a tiny bit more animated :wink:

Some English speakers have asked me already to transfer my videos into English. I think I will give it a try this year, if I find time. I will send you a message as soon as the first video is ready. Just remember ‘Echt Einfach TV’ (which means Real Simple TV).

Kind regards!

Thank you for this wonderful intuitive explanation, Kalid! You’ve done it again!

While I could follow the explanations, I did want to follow your advice and not get too hung up on an individual diagram. I also wanted to play around with the concepts, so I put together the following demos on the online Desmos calculator:

Sine/Cosine: The Dome:

Tangent/Secant: The Wall:

Cotangent/Cosecant: The Ceiling:

Visualize The Connections:

Putting these demos together and seeing the results also helped make everything clearer, and I thought others might find these useful.

Thanks again, Kalid!

hi, thanks for another great article. my only suggestion would be the large triangle in ‘visualising connections’ and the others is that it (can) look as though each label is a bit of the line and hence the total length is the sum of all the functions - i take it you mean ‘when you go to this point, you use this function to give you the length’ ie, step up from say, 1 to csc to sec NOT (1+csc+sec) = lenth of line. not a biggy but if it caught me out for a couple minutes, it might end up a road-block for someone else.

I saw this last night before going to bed, and used it this morning with my Geometry class as we began our Trig unit today.

After reading it last night, presenting it to the kids this morning, and reading through this again, trigonometry finally makes intuitive sense to me.

I am confident that this will help my students see this in a clearer light, and hopefully the handout that I put together to introduce sine and cosine today is helping them make meaningful connections.

That helps Khalid! Thanks a ton!

@kai: Sounds great, let me know and I’ll check them out.

@joe: Great feedback, I’ll see if I can add a note to clarify. When you’re making the diagrams you tend to have all sorts of unstated assumptions which aren’t there for other people :).

@Chris: That’s so awesome, I love it when the analogies come in handy for teaching. I really like how you’ve worked percentages into the worksheet, it puts a meaning behind the calculation (3/5… oh, that’s 60%!).

@Ellie: Happy to help!

This comment is to Ellie: why tangent is positive in the 3rd quadrant.

Note that tangent is NOT the slope of the circle but the slope (=rise/run=sine/cosine) of the radius extending from the center to the unit circle. When that radius is extending to the 3rd quadrant, the slope remains the same (sign and size).

Similarly, tangent is negative in the 2nd and 4th quadrants.

Kalid, you have a beautiful way of explaining things. Your illustrations, intuition buildup and Aha! moments produce a snapping feeling in my brain. Everything just falls into place never to be forgotten again.

I think that your articles are an invaluable gift to mankind. Keep it up! All the best.

Thanks Ananya, I really appreciate the encouragement! Really glad that everything clicked :).

I am confused about the ceiling diagram. How come height traversed is always 1? Seems height can be bigger as the line extends beyond the dome.

Hi vinay, try this interactive calculator for an example:

When building a ramp up to the ceiling, the distance we travel depends on the angle we pick. However, the ceiling itself is always 1 unit above the ground. (In a building the ceiling is always a constant height, no matter how steep the stairs are to get there.)

Enjoyed another door I needed to open. Thanks.

Thanks Rick, glad you enjoyed it.

Dear friend, Thank you for this precious point of view about trigonometry.
It was a pleasure reading your article. It was amazing!
Congratulations! Thanks a lot!

Thank you Rodrigo!