TRIGONOMETRY

What is trigonometry ?
Trigonometry is a branch of mathematics that studies triangles and the relationship between their sides and their angles .

Law of the sin:

This is to prove that a/sinA = b/sinB = c/sinC. First of all, in the triangle above there are 3 angles: A,B,C. And 3 sides a, b, c. The height of this triangle is h1, so we can divide this triangle in two rectangle triangles. In the left triangle,

SinA = h1/c so h1= c*sinA

Sin B= h1/a so h1=a*sinC

As we have both equations with h1, we can write:

c*sinA = a*sinC

And by cross multiplying the equation, we have

a/sinA = c/sinC

And we can use the same method to demonstrate that

a/sinA = b/sinB

and

b/sinB = c/SinC.

Therefore, we have demonstrated that

a/sinA =b/sinB =c/sinC.

Law of the cosine:

Did you know that in a triangle (see triangle above), c^2=b^2+a^2-2ab*cos(C)?

Here is the proof for it...

In the triangle ABC (above), using Pythagoras theorem, we have: c^2=h^2+(a-x)^2

which is equal to: c^2=h^2+a^2-2ax+x^2

Similarly, in the triangle ADC, using the Pythagoras theorem: b^2=h^2+x^2

Using the ratio of cosine in the triangle ADC: cosC= x/b

Which is : x=b*cosC

Eliminating x and h, we have:

c^2=b^2-x^2+a^2-2ax+x^2

c^2=a^2+b^2-2a(b*cosC)

so c^2=b^2+a^2-2ab*cosC

It works the same way for the other 2 laws which are mentioned in the picture.

So, here are our 2 proofs for the 2 interesting laws of the sin and the cosine.

PANNIR Sushmitha

JOSEPH Vanessa

2 nde euro section maths