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TRIGONOMETRY with RIGHT TRIANGLES!!!


Meaning of Trigonometry


Many people give up on learning trigonometry right after hearing the name...
"Trigono" is Greek for triangle and "metron" means measure. So basically trigonometry deals with the relationship between lengths and angles of a triangle. So now that we've got the meaning out of the way, lets start with the 3 basic functions of trigonometry.


The 3 Basic Trigonometry functions

In this article we'll only deal with trigonometry with acute angles and right triangles.
There are 3 basic trigonometry ratios :-
  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)




Before actually learning what these functions mean, we should learn some terminology.

For any acute angle θ in a right triangle:
  • The side opposite to θ is called 'Opposite' side
  • The non-hypotenuse side adjacent to θ is called 'adjacent' side
  • The longest side, that is the side opposite the right angle is called the 'hypotenuse'
*Note- Remember that the Adjacent and the Opposite sides change relative
Now that we know what all sides relative to an angle in a right triangle are called, lets define the 3 trigonometric functions.

The sine of an acute angle in a right triangle is defined as the length of the side opposite said angle over the hypotenuse.

Sin θ =   Opposite  
            Hypotenuse

The cosine of an acute angle in a right triangle is defined as the length of the adjacent side over the hypotenuse.

Cos θ =    Adjacent  
              Hypotenuse

The Tangent of an acute angle in a right triangle is defined as the length of the opposite side over the adjacent side

Tan θ =   Opposite  
               Adjacent

Do you think you wont be able to remember these 3 ratios?
well just remember the mnemonic SohCahToa
Soh - Sine is Opposite/Hypotenuse
Cah- Cosine is Adjacent/Hypotenuse
Toa- Tangent is Opposite/Adjacent

Soh,(no pun intended) now if someone asks you what the sine of 60゜ is by looking at the triangle below, this is how you would do it
you just say that as sine of an angle is Opposite/Hypotenuse and as the length of the side opposite to the 60゜ angle is √3 units and the length of the hypotenuse is 2 units.
The sine of 60゜= √3/2 and this will be true in all triangles, sin 60 will always be equal to √3/2 . This means the ratio of the opposite side to the 60゜ angle to the hypotenuse in a right triangle will always be √3/2. Lets see why...

Why are trigonometric function of an angle constant!?

To prove this we can show that sine, cosine or tangent of θ in triangles with different length of sides will be equal.
sin θ(ABC) = b/c
sin θ (DEF) = e/f
As we know that the sum of the angles in a triangle = 180゜
So, θ + 90゜ + A = 180゜
subtract 90゜ from both sides
θ + A = 90゜
A = 90゜- θ
So, now that we know that all 3 angles in ABC are equal to corresponding angles in DEF
∴ △ABC ∼ △DEF
as the triangles are similar the ratio of their corresponding sides is equal
That means, a/d = b/e = c/f
if we substitute 'k' for the constant of ratio
then, d . k = a
         f . k = c
         e . k = b

sin θ = b/c = e . k/ f . k
sin θ = b/c = e/f
Similarly we can prove this for the cosine and tangent also.


This is how far we will go into trigonometry in this article, if you have any doubts or questions related to trigonometry I'll answer them in the comments section below, in the next post, We will deal with trigonometry with general triangles and the reciprocal Trigonometric functions, Until then, Stay TUNED !   :D


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