Speaker A
Let's say we have an abstract triangle. One angle is 90 degrees, and the diagonal side has a length of one.
A concise introduction to trigonometry, covering right triangles, sine, cosine, unit circle, and other trig functions in under 10 minutes.
Ask about this video. Answers come from its transcript only — with the timestamp, so you can check them.
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Key Takeaways
- Right triangles and unit triangles are fundamental to understanding trigonometry.
- Sine and cosine functions relate triangle side lengths to angles and can be generalized using similar triangles.
- The unit circle allows trig functions to be defined for all angles, not just those under 90 degrees.
- Tangent, secant, cotangent, and cosecant functions complement sine and cosine and have unique geometric meanings.
- Trigonometry is widely applicable in real-world measurements and physics, especially with vectors.
What the video covers
- Introduction to right triangles and the significance of the 90-degree angle.
- Explanation of the unit triangle with a diagonal length of one as a foundational building block.
- Definition and naming of sine and cosine functions based on triangle side lengths and complementary angles.
- Use of similar triangles to generalize sine and cosine functions for any triangle size.
- Discussion of special angles (0, 30, 45, 60, 90 degrees) and their sine and cosine values.
- Introduction of the unit circle to extend trig functions beyond 90 degrees to 360 degrees and beyond.
- Visualization of sine and cosine as wave functions and their phase relationship.
- Explanation of other trig functions: tangent, secant, cotangent, and cosecant, and their geometric interpretations.
- Graphical behavior of tangent, cotangent, secant, and cosecant functions including asymptotes and periodicity.
- Practical applications of trigonometry in measuring heights, distances, and vectors.
Chapters
- 00:00Introduction to Right Triangles and Unit Triangle
- 00:31Sine and Cosine Functions Explained
- 00:59Generalizing with Similar Triangles
- 01:54Special Angles and Their Values
- 04:46Extending to the Unit Circle
- 05:31Sine and Cosine as Wave Functions
- 06:11Tangent and Secant Functions
- 07:25Graphs of Trig Functions and Their Properties
- 08:24Applications of Trigonometry in Real Life
Full Transcript — Download SRT & Markdown
Speaker A
That leaves two sides and two angles adjustable. Our triangle could be any one of these options.
Speaker A
Believe it or not, this niche shape is the foundation of the most useful type of geometry.
Speaker A
Trigonometry. Or just trig for short. And if you know anything about trig, you know it's all about them triangles.
Speaker A
The word literally means "three angle measure." But our triangle is a very specific type.
Speaker A
It's a right triangle. So named because 90 degree angles are called right angles. So they're the only correct angle? Give me a break.
Speaker A
No, no. This use of the word "right" is left over from a Latin term meaning "straight," because that angle makes a side that's straight up and down.
Speaker A
That's just what it's called now. Roll with it. Of course, triangles exist that aren't right.
Speaker A
But all of them can easily be divided into right triangles. So right triangles are a great building block.
Speaker A
Why did we decide the diagonal should have a length of one? Because this is a unit triangle, and units are the perfect building block.
Speaker A
Here's the thing though. For trig to do what we want it to do, it needs to relate the sides of a triangle to its angles.
Speaker A
Let's just consider one of those angles for a moment. If we change that angle but keep the diagonal the same, the other two sides change accordingly.
Speaker A
The length of each depends on the angle. In math speak, we'd say the length is a function of the angle.
Speaker A
Eventually, we'll need to know what those functions look like, but more on that later.
Speaker A
For now, we're just naming them and getting a feel for relationships. The length opposite the angle, let's call that sine.
Speaker A
S-I-N-E, sine. Why is it called that? Well... There's some debate about this. It might have been a mistranslation from Arabic to Latin, but it was probably a boob joke.
Speaker A
Humans have always been crude. There's no way to know for sure though, so believe whatever you want.
Speaker A
Okay, back to our triangle. This length is called the sine, but we also need a name for the other length.
Speaker A
It happens to be opposite the other angle, which makes it the sine of that other angle.
Speaker A
That angle is called the complementary angle, so this length is the sine of the complementary angle, or cosine for short.
Speaker A
Man, I love it when a name makes sense. Okay, cool. We have special functions for the sides of our unit triangle.
Speaker A
Now let's build bigger triangles with it. Most don't have a diagonal of one, but if the angles match, the triangles are similar.
Speaker A
Yes, that's an official math term. Similar triangles are proportional in size. Is the length of your diagonal two?
Speaker A
Well, then the other lengths are 2 sine and 2 cosine. Is the diagonal 100?
Speaker A
Well, then the other lengths are 100 sine and 100 cosine. We can write this pattern more generally.
Speaker A
It now applies to any size right triangle. And remember, you can split any triangle into two right triangles.
Speaker A
So if you're careful with your labels, this rule actually works for all triangles. Your scientific or graphing calculator knows what sine and cosine look like.
Speaker A
So if you've got that, you can do some good practical work. But let's be honest, if you've come to my channel, you're here to understand, not to do.
Speaker A
So let's go a little deeper. We've got a good handle on a few special angles.
Speaker A
Zero, 30, 45, 60, and 90 degrees. Let's start with 45 degrees. In this case, the angles are equal, which means the sides are also equal.
Speaker A
Pythagorean theorem gives us the number. Like I said, we've got a good handle on these. They're the low-hanging fruit.
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At 30 degrees, the opposite side is half the diagonal. We know this because two of them form an equilateral triangle.
Speaker A
Never forget, these right triangles are building blocks. Anyway, Pythagorean theorem gives us the other number.
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At 0 degrees, we take this to the extreme. The opposite side is clearly zero, which makes the diagonal and the bottom sides the same.
Speaker A
The 60 and 90 degree cases we get from symmetry. They're the opposite of 30 and 0 degrees, respectively.
Speaker A
These angles we know very well, but the rest aren't so obvious. What are the values for sine and cosine at, I don't know, 33 degrees or 72 degrees?
Speaker A
To figure that out, we need to go all the way around. This doesn't just stop at 90 degrees.
Speaker A
It goes to 360 degrees and keeps going. The outer corner of this unit triangle draws an entire circle, which we call the unit circle.
Speaker A
This circle is simply the set of all unit triangles. That's it. Traditionally, it only shows numbers for a few special cases, but it encompasses every angle.
Speaker A
And we can use it to figure out what sine and cosine actually look like.
Speaker A
Sine might be this side of the triangle, but it's also this length over here.
Speaker A
They're the same because rectangles. Now, let the angle go around at a steady rate.
Speaker A
If that sine length were to draw on a scrolling piece of paper, out pops a wave.
Speaker A
This is what the sine function looks like. We can do the same thing for cosine.
Speaker A
We just need the paper to scroll up and down. But if we superimpose them, we can compare.
Speaker A
It turns out sine and cosine are the same function, just 90 degrees apart. Which makes sense because that's how we defined cosine.
Speaker A
It's the sine of the complementary angle. It is a sine function. That just leaves the other trig functions.
Speaker A
Tangent, secant, cotangent, and cosecant. All four of them are on the unit circle too.
Speaker A
They're just not necessarily inside the circle. A tangent line touches a shape at one point.
Speaker A
Tangent literally means touching. The tangent function is a segment of the tangent line. It goes from the outer corner of the triangle to the horizontal axis.
Speaker A
Because that's the part of the line within the angle. Just like sine and cosine, this changes as the angle changes.
Speaker A
Which makes sense. It happens to be equal to sine divided by cosine. For secant, we're going to have to think a little more outside the box. I mean circle.
Speaker A
This tangent forms a larger right triangle with the horizontal axis. The length along that horizontal axis is the secant.
Speaker A
Secant literally means to cut. This is the line that cuts off the tangent to give it the length it has.
Speaker A
It's literally cutting through the tangent line. And you can see it also depends on the angle.
Speaker A
We get cotangent and cosecant the same way we got cosine. They're the complementary versions.
Speaker A
So they're simply found on the other side of the diagonal. Now that we have every modern-day trig function on the diagram, we can see how they all relate to each other.
Speaker A
And how they all change as the angle changes. So what do all these functions look like on a graph?
Speaker A
They're not as pretty as sine and cosine, but we can still figure them out.
Speaker A
The unit circle shows us that tangent goes to infinity in a couple of places.
Speaker A
And has a value of zero at zero degrees. That looks something like this. And that pattern repeats forever in both directions.
Speaker A
Cotangent is the complementary tangent. So it's just mirrored and a little shifted, but basically the same.
Speaker A
We can do a similar thing with secant. It looks like this. The cosecant is the complementary secant. So again, it's just mirrored and a little shifted.
Speaker A
What I like most about these two is how they compare with sine and cosine.
Speaker A
Kind of beautiful, isn't it? Okay, but how is this the most useful type of geometry?
Speaker A
Triangles are the strongest shape, which is why we use them in bridges and other support structures.
Speaker A
Speaking of which, please consider supporting my work. It'll get you access to exclusive content, links in the doobly-doo.
Speaker A
These triangles don't have to be tangible though. Say you want to measure the height of a building.
Speaker A
Could you make a really long tape measure? Sure. But why not just measure the easier length along the ground and imagine a triangle?
Speaker A
The same goes for measuring the distance across a river. And if you're doing physics, you're probably working with vectors.
Speaker A
Got a vector that doesn't line up with your coordinates? You're going to need a triangle.
Speaker A
Triangles are everywhere. They're th
Speaker A
I think you're just getting in your head about it. I want to make sure I'm enunciating the word diagonal.
Speaker A
If anything, you are over... Diagonally! I feel like we're going to end up with penwings here in a second.
Speaker A
Right! Penwings. Mirrored. Mirror. Mirrored. Mirrored. Secant? More like see-can. I'm such a nerd.
Topics:trigonometryright triangleunit circlesinecosinetangentsecantcotangentcosecantmath tutorial
