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 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


You Might Also Like :- 


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

Comments

Popular Posts

Cathode Ray Experiment

This experiment was conducted by J.J. Thomson (Sir Joseph John Thomson) in the year 1897. This experiment proved that atom is made up of fundamental particles which are much smaller than the smallest atom 'hydrogen' This experiment helped to discover electron. According to J.J. Thomson, the cathode rays consisted of very light, small and negatively charged particles. He named the particles "corpuscles" which were later known as electrons

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.

Rutherford Alpha Ray Scattering Experiment

Rutherford Alpha Ray Scattering Experiment Hey, Guys, most of you might have heard about the alpha ray scattering experiment and if you want to know in detail about Rutherford's model and the experiment he conducted, this is the right place for you... But first: Things You Must Know Some basic information that will help you understand rutherford experiment properly: Proton is a sub-atomic particle which is positively charged and has a mass of 1u. Alpha particles are helium atom with a charge of +2 as they have lost 2 electrons. Alpha particles have an atomic mass  of 4u. Gold is highly malleable and can be beaten into very thin sheets. Experiment Rutherford conducted his experiment in the following way: Rutherford took a very thin gold foil and bombarded it with high energy alpha particles. He placed a layer of zinc sulphide on the walls where the experiment was taking place because when alpha particles strike zinc sulphide layer, it results i

Important Mathematical Constants!

Important Mathematical Constants Mathematical constants are those numbers that are special and interesting because they come up in the various fields of mathematics like geometry, calculus etc. These mathematical constants are usually named after the person who discovered it and they are represented by a symbol that is usually picked up from the Greek alphabet. Mathematical constants are by definition very important. In this article we will take a look at certain mathematical constants that are more commonplace than others. 1.       π (pi) or Archimedes constant (~3.14159):   π is defined as the ratio of the circumference of a circle to its diameter. This is probably the most popular mathematical constant. So π is the circumference of the circle whose diameter is 1 unit. You might have seen it popping up when calculating the area of a circle (πr 2 ) or the circumference of a circle (2πr). It has many uses throughout mathematics from calculating the area of certain shap

Paid Apps For Free ??

Everyone wants to play a game like GTA on iPad, it is easy on a computer to download such games but difficult on smart devices like iPad, tablet,  or smartphones. We can buy them but not everyone can buy games. But no worries guys there is a solution to this problem where one can have fun of playing games without spending their precious money .(underlined apps have downloading links  given at the end) iOS There many apps through which you can download these paid games for free. Also, there are many sites for the same. One of the most helpful apps is a Chinese app.It is called haimawan. If on a ios device, then you just have to click install which will redirect to settings  add a profile  and boom enjoy it as you wish . but it does not always work, it might work for few days and then the verification problem which may not cure. One more app is tutu which is a fantastic app and always work (99.9% sure .. nothing is perfect). X cross was the perfect app but is closed now an

2-D & 3-D GEOMETRY

2-D & 3-D GEOMETRY We all have some amount of geometry. We know that any line can be represented on the Cartesian plane. Any figure can be drawn on it. But can we represent a 3-d object on it. Yes we can. A Cartesian plane has 2 axis. While representing in 3-D we need to add a third axis. This axis does not come in between the axis or in the same plane. It appears to be coming out of the paper as we cannot represent a 3-d object on a 2-d surface. This new z-axis represents a line coming out of the screen. Before understanding 3-d geometry you need to imagine this axis coming out of the screen.  REMEMBER : all the three axis are perpendicular .i.e there an angle 0f 90 between them and they meet at the origin If you are unable to imagine you can take a thick book as an example. Any corner becomes it origin and the three edges as the three axis REPRESENTING 3-D GEOMETRY Like in 2-d geometry we represent the value of the different axis as (x,y) we use the same m

Animal and Plant Cells

 Cells Cells are the basic functional, biological and structural unit of life. The word cell is a Latin word meaning ‘small room’. Cells are also known as building blocks of life.  The branch of science that deals with the form, structure, and composition of a cell is known as Cytology. All organisms around us are made up of cells. Bacteria, ameba, paramecium, algae, fungi, plants and animals are made up of cells.  Cells together form tissues. And tissue together makes an organ. History Of Cell The cell was discovered by Robert Hooke in 1665. He assembled a simple microscope and observed a very thin slice of cork under his primitive microscope. The cork was obtained from the outer covering of a tree called bark. Robert Hooke observed many little-partitioned boxes or compartments in the cork slice. These boxes appeared like a honey-comb. He termed these boxes as the cell. He also noticed that one box was separated from another by a wall. What Ho