Skip to main content

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 Î¸ 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 Î¸ =    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


You Might Also Like :- 


Enjoy your high school with - High School Pedia : www.highschoolpedia.com

Comments

Popular Posts

Piezoelectric Crystal

Piezoelectric Crystal {literally_pressing electricity} Ever wondered what it might be like to live in dark or sleep in a hot summer day without an AC ? Absurd, isn't it ? But with the rate at which we are using our resources, it may not be long before it actually happens. Scientist across the globe are searching for new resources, stocks and technology to prevent the near, unpleasant future of current and the next gen. One of these, an unexploited tech is that of Piezoelectricity. Piezoelectricity is an ability of some special materials (like crystals of quartz, ceramics of lithium niobite, gallium arsenide, zinc oxide) to trigger an electric charge when supposed to mechanical stress. These materials are available in basic instruments that we use in our daily life like microphones, quartz watch, gramophones. All these materials are run by the piezoelectric current. But how does it work? When you apply force (mechanical force) on it, an electric potential, on eit

The Inverse & Reciprocal TRIGONOMETRIC Functions

So, this is my second post on trigonometry. In this post we're gonna cover the reciprocal and the inverse Trigonometric functions. If you haven't seen my first post you should definitely view it as it covers the basics of Trigonometry The Reciprocal Trigonometric Functions The reciprocal Trigonometric function of Sine is Cosecant, of Cosine is Secant & for Tangent it is Cotangent. Cosecant (Csc θ = 1/Sin θ) or (Hypotenuse/Opposite) Secant (Sec θ = 1/Cos θ) or (Hypotenuse/Adjacent) Cotangent (Cot θ = 1/Tan θ) or (Adjacent/Opposite) We can also represent Tan θ in another way. As Tan θ = opposite/adjacent  & Sin θ = opposite/hypotenuse  & Cos θ = adjacent/hypotenuse ∴ Tan θ = Sin θ/Cos θ (The hypotenuses cancel out) As Cot θ = 1/Tan θ  So, we can also represent Cot θ as Cos θ/Sin θ.

Anode Ray Experiment

→Anode ray experiment was conducted by E Goldstein. →These rays are also known as canal rays. →This experiment helped in the discovery of the proton. Apparatus Used A discharge tube  was taken in which there were 2 electrodes i.e. Anode(+ve) and the cathode (-ve). The tube was filled with an inert gas. A perforated or porous cathode was used. A layer of zinc sulphide was placed at the back of the cathode. There was a vacuum pump in the tube. High voltage (5000v-10000v) was allowed to flow through the system. It was observed that when the gas was at 1atm(atmospheric pressure ) no change was seen in the tube.  When the   pressure   was decreased inside the tube, a glow could be seen at the back side of the cathode.

High School Pedia

It is an initiative by some students to spread the light of knowledge to everyone and everywhere. It was started in the year 2015 and has grown rapidly in the past few months. By the means of this website, we try to provide information on every topic that we can reach up to. You can find different articles on this website. All these articles are written in simple language so that everyone can understand it and learn from it. We at High School Pedia believe in creative learning and this is the reason why we add our own edited graphical representations in every article. Once a very learned man said, “Knowledge increases by not keeping it to yourself but by sharing it with others”. And we follow the same motto “Share to Learn”. The team of High School Pedia tries its best to provide you with the best and original content. Unlike many other websites, High School Pedia is famous for its original and inspiring content.

Levitation 2

LEVITATION II To be completely honest I was going to start this with a pun. I did think of one but it doesn’t float… I am sorry I just had to. Anyway, this is the second part to the article on super cool ways of making things levitate. Go check the first part out if you haven’t already. Actually, the first part may have become repulsive with all the magnets and stuff, but I promise this will be more attractive. Get it? No? I’ll stop now. I am just going to jump straight into it. 1.    Electrostatic Levitation I know you are probably sick and tired of magnets but they are the best way you know… This method is somewhat similar. You remember that cool science experiment you did with two straws attracting or repulsing each other based on their charge? So basically using the same principle we can make a charged object levitate. But before you try it, let me tell you it won’t be easy. Even impossible according to our Mr. Earnshaw. He even made a law (the law is

Isotopes, Isobars and Isotones

Isotopes These are elements which have the same atomic number but different atomic mass . They have the same atomic number because the number of protons that are inside their nuclei remains the same. But, they have different atomic mass because the number of neutrons that are also inside their nuclei is different. As the number of protons inside nuclei remains same, therefore the overall charge of the elements also remains same as in isotopes: no of protons = no of electrons . Hence, as isotopes overall charge remains neutral, therefore their chemical properties will also remain identical.   Therefore, Isotopes are chemically same but physically different.

Upcoming Marvel Movies

Hey, guys! Today we are going to talk about the 5 most awaited and upcoming marvel movies. Everyone is eagerly waiting right ?!            1) GUARDIANS OF THE GALAXY vol.2 (2017) Last month marvel shared the first teaser of Guardians of the Galaxy vol. 2   describing it as the official sneak-peak. All five main characters- star lord, gamora, rocket, Drax and 'baby' groot . It is set two months after the first movie where the ravagers find out that peter quill double crossed them about the infinity stone (orb). That's all. Everything else is yet to be discovered.