একক বৃত্তের ত্রিকোণমিতি এবং ত্রিকোণমিতিক ফাংশনের পর্যাবৃত্ত ধর্ম

The Unit Circle

So far we have only defined the trig functions on angles that are positive and strictly less than $90$ degrees. If you 'played' with your calculator you will notice that the calculator and calculate sines and cosines and tangents of angles larger than 90 degrees. Why does the calculator output "Math Error" when you try calculating the Tangent of $90$? Why is $\sin 90^\circ = 1$?

Imagine a circle on the cartesian co-ordinate plane centred at the origin with radius $1$.

A Ray that starts from the origin is drawn, and the line makes an angle $\theta$ with the positive $x$-axis. Call the intersection of the ray and the circle $P$ with co-ordinates $(x,y)$.

Now, the origin, point $P$, and the point $(x,0)$ makes a right-angled triangle.

So, now we have a new definition of the trig functions!

In the unit circle, $\sin \theta$ is the $y$ co-ordinate of point $P$, $\cos \theta$ is the $x$ co-ordinate of point $P$, and $\tan \theta$ is $\frac yx$ where $x$, $y$ are co-ordinates of point $P$

A Unit Circle is exactly what it sounds like: a circle with radius $1$ unit. It is located at the origin, the intersection of the x and y axis:

You may be wondering what is the use of this circle, and how it relates to trigonometry. To see that, let's draw a triangle.

In the diagram,

We know that $\sin \theta=\dfrac{PH}{PO}$. However, from the property that the radius of a unit circle is $1$, we see that $PO=1$. Thus, $\sin \theta=PH$. Since $PH$ is also the $y$ coordinate of point $P$, we have that $P=(x,\sin\theta)$.

Similarly to $\sin$, you can find that $\cos \theta= OH$. Since $OH$ is also the $x$ coordinate of point $P$, we have that $P=(\cos\theta,\sin\theta)$

What about $\tan \theta$?

Can we simplify this any further? We don't know the lengths of either $PH$ nor $OH$, so perhaps we can't. However, there is a clever geometrical interpretation: $\tan \theta$ is simply the slope of line $PO$! (Make sure this makes sense.)

Now we've seen how the unit circle can simplify the definitions of $\sin$ and $\cos$ a lot.

After seeing the intuition of using parts of the unit circle as $\sin\theta$ and $\cos\theta$, I should introduce you to the Unit Circle Definition of Trig Functions.

Suppose we have a unit circle, and a point $P$ on the outside perimeter of it. The angle that the line through $P$ and the origin $O$ makes with the x-axis is $\theta$. We define the $x$ coordinate of $P$ to be $\cos\theta$ and the $y$ coordinate of $P$ to be $\sin\theta$ for all $\theta$, not just acute $\theta$. (Contributor Note: Someone who is proficient with the plotting tools should clean up this diagram and all the ones below.)

Why is this new definition useful? For one, it applies to all possible real values of $\theta$, even negative ones and ones where $\theta > 360^{\circ}$! We certainly couldn't have done that with our right triangle trigonometry.

It also makes a few identities extremely obvious. For example, the identity $\sin \theta=\sin (180^{\circ}-\theta)$.

Looking at the diagram, this is clearly true:

Another example is the famous $\sin^2\theta+\cos^2\theta=1$. From the unit circle, we see that any angle creates a right triangle, and so by the Pythagorean theorem we have that $\sin^2\theta+\cos^2\theta=1^2=1$, thus proved.

Congratulations! You have learned what the Unit Circle is, how it relates to trigonometric functions, and the usefulness of this new interpretation.

Click for Extras

Extras

We can also express all the other trigonometric functions on the unit circle too. Let's first do $\tan \theta$.

The labelled segment has length $\tan\theta$. This can be seen using similar triangles. If the length of the segment labelled $\tan\theta$ is $x$, then by similar triangles, $$\dfrac{\sin\theta}{\cos\theta}=\dfrac{x}{1}$$ so $x=\tan\theta$.

We can also label the values of $\csc\theta$, $\sec\theta$, and $\cot\theta$ in a similar way:

See if you can mimic my proof of $\tan\theta$ to prove that the other lengths indeed represent the trig functions.

Hi guys, I'm back. So it's high time for some algebra and trig :D

Okay, let's look at the equation of the unit circle: $$x^2 + y^2 = 1.$$

Now, forget that we were even talking about the unit circle. What does that equation I just mentioned remind you of? [please don't say the unit circle :P ]

Yep, it's something derived from the Pythagorean theorem. It also has to do with trig. I'm gonna build up the suspense further…

This is a well-known trigonometric identity. But how was it derived? (yes, I was forshadowing something at the beginning of this post…)

So here's the unit circle with one of its radii drawn. You see the right triangle with side lengths $x$, $y$, and $1$. Notice the marked angle $\theta$ made by the radius and the $x$-axis.

Now we can do substitutions between the unit circle and this awesome trig identity. Here's how it works:

We'll cap this post of with a diagram that sums all this info up:

Here you go!

So, the takeaway lesson is this:

Have fun!