Trignometry and Circles. Are they related?

 The Tangent and Secant. Circle and Trigonometry .

One is related to a closed figure with the highest number of sides ( basically infinite sides ) and the other is about a closed figure with the least number of sides . At first glance, we don't see any correlation between these two concepts, but they have the same terms expressing their important topics. 

Most people might think that the mathematicians of that time just ran out of words to name the innumerable things they discover but that is not the case in this situation. 

The Tangent and Secant of a circle and that from trignometry have a cardinal relation, which we will dig deep and uncover in this blog, 

Let's consider a circle with centre O and radius 1 cm ( for making calculations easier) .Hence OA and OB are radii of the circle ( Both of 1 cm). Next, draw a line segment perpendicular from B to OA and name this point C. Hence BC is perpendicular to OA.

Now, OBC is a right angled triangle.

Considering the trigonometric functions of this triangle with respect to angle O :-

(By opposite, we mean the side opposite to the angle O and Hypotenuse is the longest side of the triangle) 

But, we know that OB = 1 cm ( as it is the radius)

Hence sin O = BC 

Similarily 

(Adjacent means the side adjacent to the angle O) 

and as OB = 1 cm

cos O = OC 

Next, we need to draw a line touching the circle only at A ( Basically, a tangent through A) and extend radius OB to touch the tangent drawn at D. One thing we need to keep in mind is that a tangent is always perpendicular to the radius drawn from the centre to the point of contact. ( Wondering Why? Click here! )

From the diagram in the right, we see that ∠ O is common and  ∠ C and  ∠ A are equal , each being 90 degrees

Δ  OBC and  Δ  ODA are similar  by AA Criteria,

Δ OBC ~ Δ ODA

In  Δ ODA, which is right angled

And we know that, OA is 1cm . Hence tan O = DA

Due to similarity of the triangles 

Hence ,

So DA which is the tangent of the circle is actually also the tangent of the angle subtended at centre O!

We can conclude that:-

The length of the line segment tangent to a circle that connects to a central angle, is the tangent of the central angle!

Now, We  also see that OD in the above figure is actually a secant ( once extended , it will cut the circle at two different points) and it is also the hypotenuse of the bigger triangle ODA.

Using the same similar triangles, We can infer that

Hence, OD = 1/ cos O ( OA = OC = 1cm as they are the radii)

OD = secant O

We can also conclude that The mathematicians of the past were not so run out of latin words after all ( Sorry Archimedes for blaming you 😝) and everything they did had a reason behind it.

It is also fascinating how every tiny detail in Mathematics are inter- related and two seemingly uncorrelated topics are actualy sister-consent  ( or brother-consent)  to each other! 

Provocative right? If this blog kindled that intriguing nature inside you, Don't worry, We will have more maths blogs to quench that thirst of curiosity inside you! So keep your eyes peeled for new blogs!