Welcome to Professor MAD.
Mathematics for Everyone.
Today we are going to talk about the basics of trigonometry.
Trigonometry is one of the most important branches of mathematics,
that finds vast application in diverse fields.
Trigonometry simply means calculations with triangles.
That's where the "Tri" comes from.
Especially it deals with the right angled triangles,
which contain a 90 degree angle.
In trigonometry,
we study the relationship between the sides and angles of the
right angle triangle.
Trigonometry skills are essential in various careers,
including Architecture, Astronomy, Geography, Engineering and many more.
First let's talk about right angled triangles.
A right angled triangle is a triangle with one of the angles at 90 degrees.
We indicate it by using a little box sign.
The side opposite the right angle does the longest, referred to as the hypotenuse.
The other two sides are labeled concerning the reference angle.
The opposite side is across from a given angle.
For example,
if you take Alpha as the reference angle the opposite side is AB.
If you take beta as the reference angle, the opposite side is CB.
The adjacent side is the non-hypotenuse side next to a given angle.
For example,
if you take Alpha as the reference angle the adjacent side is CB.
If you take beta as the reference angle, the adjacent side is AB.
Knowing how to accurately label the sides for a specific reference angle is essential.
As I mentioned earlier,
the triangle is the simplest geometric shape
which only has three sides and three angles.
In right angled triangles, we surely know one is always a right angle.
So there are five unknowns.
If we know any two of these,
We can calculate others using the knowledge of trigonometry.
Before going further,
we have to brush up on our knowledge of geometry.
The first one is that,
the sum of all the interior angles of a triangle is 180 degrees.
We call this the "Triangle Sum Theorem".
So if we consider two right angled triangles,
the sum of the interior angles of each triangle is equal to 180 degrees.
And we already know that in right angle triangles, one angle is always 90 degrees.
So the sum of the rest two angles is also equal to 90 degrees.
Now assume the angles A and Q are also equal.
Let's say those are equal to Alpha degrees.
Then both B and P are equal to 90 minus Alpha.
Let's say it's equivalent to Beta degrees.
Now we can see in both triangles,
the corresponding angles are equal to each other.
so those are similar triangles.
Early mathematicians have identified that,
in similar triangles corresponding sides are always in the same ratio.
Now we have a relationship between 2 similar right angled triangles.
This relationship can be separated into three parts.
Let's rearrange these three equations.
Are you with me up to this point?
So let's clear the rest and keep our last equations.
The ratio between two sides of a triangle,
equals the ratio between corresponding sides of similar triangles.
This is very useful.
It is the foundation of trigonometry.
So in trigonometry, we name those ratios according to the reference angle.
Let's consider the angle Alpha as our reference angle.
c and r are hypotenuses.
a and q are opposite sides for angle Alpha.
p and b are adjacent sides for angle Alpha.
So the first relationship is the ratio
between the opposite and hypotenuse of the alpha angle.
The second relationship is the ratio between the
adjacent and hypotenuse of the alpha angle.
The third relationship is the ratio between
the opposite and adjacent of the alpha angle of a right angled triangle.
For any right angled triangle with the same reference angle,
the values of the ratios do not change
We name those sine, cosine and tangent ratios.
Shortly sin, cos and tan.
To indicate the reference angle we wrote those like this.
Those are the three fundamental trigonometric ratios.
So next time someone mentions a sin, cos or tan,
don't be afraid.
They are talking about the ratio between two sides of a right angled triangle,
in reference to an angle.
For any particular angle there is a constant value for these ratios.
For example let's consider the angle is equal to 30 degrees.
For the sine ratio we get 0.5.
For the cos ratio we get 0.866.
For the tan ratio we get 0.577.
These values are the same for any right angled triangle with a 30 degree angle.
No matter what the size and orientation are.
If we consider the alpha as 45 degrees,
For the sine ratio we get 0.707.
For the cos ratio we get 0.7072.
For the tan ratio we get 1.
Likewise every angle has constant sin cos and tan values.
If you have any doubt about this,
please try to draw several triangles at different sizes but with the same angles.
Then measure the lengths and calculate the ratios of the corresponding sides of each triangle.
Each triangle will get the constant sin, cos and tan values
common to every triangle.
But we don't have to measure the sides and
calculate these ratios every time we need them.
There are pre-calculated value tables for those
and most of our calculators come with trigonometric functions.
Let's talk about the Practical use of trigonometry.
Here we have a flag pole.
You have to measure its height.
It's not easy to measure it from the ground to the top.
Instead we can measure
a certain distance from the pole along the land
and the upward angle to the top of the tower.
To measure the upward angle,
we can use a simple Astrolabe .
Now we know the angle and the adjacent side of that angle.
We have to find the opposite side.
Here we can simply use the tan ratio.
Since we can find out the tan value for the measured Alpha angle
using the trigonometric table,
we can easily calculate the height of the pole.
Likewise trigonometry is very useful in many practical applications
It's essential to memorize the names and ratios correctly.
Sin is the opposite divided by the hypotenuse.
Cos is the adjacent divided by the hypotenuse.
Tan is the opposite divided by the adjacent.
There is a simple mnemonic.
Take the first letter of each term and arrange it like this.
It sounds like SOHCAHTOA.
That's all for today.
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