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Trigonometric Ratios

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05 September 2020                

Read time: 10 minutes

Introduction

Trigonometry emerged during the 3rd century BC from applications of geometry to astronomical studies. The Greeks used it for the calculation of chords, and Indian mathematicians created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.
Throughout history, trigonometry has been applied in areas such as geodesy, surveying, astronomy, celestial mechanics, and navigation.
Trigonometry is known for its many identities which are equations used for rewriting trigonometric expressions to solve equations, to find a more useful expression, or to discover new relationships.

trigonometry origin words

Trigonometry deals with right-angled triangles, where one of the internal angles is 90°. Trigonometry is a system that helps us to work out missing or unknown side lengths or angles in a triangle.
 



What are Trigonometric Ratios?

The trigonometric ratios of a triangle are also known as trigonometric functions. These are real functions that relate an angle of a right-angled triangle to ratios of a Right-angled triangle. They are widely used in all sciences that are related to geometry and even have applications in engineering, navigation, architecture, geodesy, and many others. 
The most widely used trigonometric functions are the sine, cosine and tangent are 3 important trigonometric functions and are abbreviated as sin, cos, and tan. Their reciprocals are cosecant or cosec or csc, secant or sec, and cotangent or cotan or cot respectively, which are less used in modern mathematics.
Let us see how these ratios or functions, evaluated in the case of a right-angled triangle.
Consider a right-angled triangle, where the longest side is called the hypotenuse, and the sides opposite to the hypotenuse are referred to as the adjacent and opposite sides.

Triangle of trigonometry

 

 

Ratio Name Formulas for Angle Theta Reciprocal Identities
 Sine or Sin theta  sin θ = Opposite Side/Hypotenuse            sin θ = 1/cosec θ
 Cosine or Cos theta  cos θ = Adjacent Side/Hypotenuse  cos θ = 1/sec θ
 Tangent or Tan theta  tan θ = Opposite Side/Adjacent  tan θ = 1/cot θ
 Cotangent or Cotan or Cot theta            cot θ = Adjacent Side/Opposite  cot θ = 1/tan θ
 Secant or Sec theta  sec θ = Hypotenuse/Adjacent Side  sec θ = 1/cos θ
 Cosecant or Csec theta or Csc theta            cosec θ = Hypotenuse/Opposite  cosec θ = 1/sin θ

 

The Angle of ElevationTrigonometric ratios are frequently expressed as decimal approximations.

Angle of elevation

 

In this picture, the highlighted angle denotes the angle of elevation of the top of the tree as seen from a point on the ground.
The angle of elevation is always measured up from the ground. It is defined as an angle between the horizontal plane and oblique line from the observer’s eye to some object above his eye. It is always inside the triangle. Eventually, this angle is formed above the surface. 
You can think of the angle of elevation about the movement of your eyes. You are looking straight ahead and you must raise (elevate) your eyes to see the top of a tree or think of an elevator that only goes up

The angle of Elevation Formula
Tan θ = Opposite Side/Adjacent Side


The Angle of Elevation

Angle of depression

In this diagram, highlighted denotes the angle of depression of the boat at sea from the top of the lighthouse. The angle of depression is always OUTSIDE the triangle.
The angle of depression is just the opposite scenario of the angle of elevation. It is defined as if the object is kept below the eye level of the observer, then the angles formed between the horizontal line and the observer’s line of sight is called the angle of depression.

You can think of the angle of depression about the movement of your eyes. You are standing at the top of the lighthouse and you are looking straight ahead. You must lower (depress) your eyes to see the boat in the water.

elevator and elevation angle

The angle of depression Formula
Tan θ = Opposite Side/Adjacent Side
Note:  Angle of Elevation = Angle of depression
 



Basic Properties of Trigonometric Ratios

Two angles are supplementary to each other if their sum is equal to 180°.
Trigonometric-ratios of supplementary angles are given below

sin (180° - θ) = - sin θ

sin (180° + θ)  =  - sin θ

cos (180° - θ) = - cos θ

cos (180° + θ)  =  - cos θ

tan (180° - θ) = - tan θ

tan (180° + θ)  =  tan θ

sec (180° - θ) = - sec θ

csc (180° + θ)  =  - csc θ

csc (180° - θ) = -csc θ

sec (180° + θ)  =  - sec θ

cot (180° - θ) = - cot θ

cot (180° + θ)  =  cot θ

Let us see how the trigonometric ratios of supplementary angles are determined. To know that, first we have to understand the ASTC formula. The ASTC formula can be remembered easily using the following phrases.
 

All Silver Tea Cups or All Students Take Calculus
ASTC formula has been explained clearly in the figure given below.

astc formula

More clearly 

quadrants and ratios

From the above picture, it is very clear that 
(i)  (180° - θ) falls in the second quadrant and 
(ii)  (180° + θ) falls in the third quadrant

In the second quadrant (180° - θ), sin and csc are positive and other trigonometric ratios are negative.
In the third quadrant (180° + θ), tan and cot are positive and other trigonometric ratios are negative.
 



Conversion of Trigonometric Ratios

Using fundamental identities, we can convert every Trigonometric ratios in terms of the other
Like if Sin is to be converted to other ratios, we can do as follows…
Since
         Sin2θ+Cos2θ=1                       (1st identity)
         => Sin2θ = 1 - Cos2θ
         => Sin θ = √(1-Cos2θ) ……… (1)
         This way Sin θ has been converted into Cos θ,

Now let’s convert Cos θ to another form
Let’s replace Cos θ by another reciprocal identity 
Cos θ = 1/Sec θ 
 then replace Sec θ with √(tan2 θ +1)
          =>Cos θ = 1/√(tan2 θ +1)
          => Cos2 θ = 1/tan2 θ +1)
Now substitute this value in identity….. (1) to convert Sin θ into tangent form.
Hence, this way we can interconvert Trigonometric ratios.
 
Following Table represents all Trigonometric Conversions:

identity conversion table

 

Important Conversions
When we have the angles 90° and 270° in the trigonometric ratios in the form of

  • (90° + θ)
  • (90° - θ)
  • (270° + θ)
  • (270° - θ)

We have to do the following conversions, 
sin θ <------> cos θ
tan θ <------> cot θ
csc θ <------> sec θ

For example,
sin (270° + θ)  =  - cos θ
 cos (90° - θ)  =   sin θ


For angles 0° or 360° and  180°, we should not make the below conversions. 
Evaluation of Trigonometric Ratios Using ASTC Formula

 

1. sin (180° - θ)

To evaluate sin (180° - θ), we have to consider the following important points. 
(i)  (180° - θ) will fall in the IInd quadrant. 
(ii)  When we have 180°, "sin" will not be changed as "cos"
(iii)  In the IInd quadrant, the sign of "sin" is positive. 
Considering the above points, we have 
sin (180° - θ)  =  sin θ

 

2. cos (180° - θ)

 To evaluate cos (180° - θ), we have to consider the following important points. 
(i)  (180° - θ) will fall in the IInd quadrant. 
(ii)  When we have 180°, "cos" will not be changed as "sin"
(iii)  In the IInd quadrant, the sign of "cos" is negative. 
Considering the above points, we have  
cos (180° - θ)  =  - cos θ
 

Trigonometric Ratios of Complementary Angles

Two acute angles are complementary to each other if their sum is equal to 90°. In a right triangle the sum of the two acute angles is equal to 90°. So, the two acute angles of a right triangle are always complementary to each other.
Let ABC be a right triangle, right-angled at B.  

complementary triangle

If  ∠ACB = θ, then  ∠BAC = 90° - θ and hence the angles  ∠BAC and  ∠ACB are complementary.

1 . For the angle θ

2. For the angle (90° - θ)

Comparing (1) and (2) we get    Trigonometric Ratios of Complementary Angles

sin θ =  AB/AC

cos θ  =  BC/AC

tan θ  =  AB/BC

cosec θ =  AC/AB

sec θ  =  AC/BC

cot θ  =  BC/AB

  cos (90 - θ) =  AB/AC

sin (90 - θ)  =  BC/AC

tan (90 - θ)  =  BC/AB

cosec (90 - θ)  =  AC/AB

sec (90 - θ)  =  AC/BC

cot (90 - θ)  =  AB/BC

sin θ  =  AB/AC  =  cos (90 - θ)

cos θ  =  BC/AC  =  sin (90 - θ)

tan θ  =  AB/BC  =  cot (90 - θ)

cosec θ  =  AC/AB  =  sec (90 - θ)

sec θ  =  AC/BC  =  cosec (90 - θ)

cot θ  =  BC/AB  =  tan (90 - θ)

 

Example 1

 

 

Evaluate  cos 56° / sin 34°

Solution: Angles 56° and 34° are complementary. Use trigonometric ratios of complementary angles. 
                              cos 56° / sin 34°  =  cos 56° / cos (90° - 34°)
                              cos 56° / sin 34°  =  cos 56° / cos 56°
 

cos 56° / sin 34°  =  1

 

Example 2

 

 

Evaluate tan 25° / cot 65°

Solution: Angles 25° and 65° are complementary. Use trigonometric ratios of complementary angles.
                               tan 25° / cot 65°  =  tan 25) / tan (90° - 65°)
                              tan 25° / cot 65°  =  tan 25° / tan 25°
 

tan 25° / cot 65°  =  1



How to learn Trigonometric Ratios?

SOH CAH TOA: an easy way to remember trig ratios
The mnemonic sohcahtoa can be used to help us remember the definitions of the 3 trigonometric functions sine, cosine, and tangent. Here's how it works:

Acronym Verbal Description Trigonometric Formula
 SOH  sine is opposite over hypotenuse  sine theta = opposite/hypotenuse
 CAH  cos is adjacent over hypotenuse  cos theta = adjacent/hypotenuse
 TOA  tan is opposite over adjacent  tan theta = opposite over adjacent

 

For example, if we want to recall the definition of the sine, we reference 
SOH since sine starts with the letter S. The O and the H help us to remember that sine is opposite over hypotenuse. This way SOH CAH TOA or SOHCAHTOA can be used to remember the ratios, using which the other three can be computed further. 
 

Example 1

 

 

What is the tangent of 30°?

sohcahtoa example

Solution: Sine is defined as the ratio of the opposite to the hypotenuse (SOH) (from soh cah toa)

sohcaotoa solution 

sin(A)   =   opposite / hypotenuse
             =    AB / BC  
             =     5 / 3



Applications of Trigonometric Ratios in real life

While it may seem as if trigonometry is never used outside of the classroom, you may be surprised to learn just how often trigonometry and its applications are encountered in the real world.

1. Real-Life Application of the Pythagorean Theorem

The Pythagorean Theorem is a statement in geometry that shows the relationship between the lengths of the sides of a right triangle – a triangle with one 90-degree angle. The right triangle equation is a2 + b2 = c2. Since one is able to find the length of sides, given the lengths of the two other sides makes the Pythagorean Theorem a useful technique for construction and navigation.

2. Architecture and Engineering

A huge section of architecture and engineering relies on triangular supports. Trigonometry plays a major role in engineering as it helps to calculate the correct angles to determine the length of cables, the height of support towers, and the angle between the two when gauging weight loads and bridge strength. Trigonometry helps builders to correctly layout a curved wall, figure the proper slope of a roof or the correct height, and rise of a stairway. Trigonometry can be used to determine the square footage of a curved piece of land.

trigo application arch and eng


4. Electrical Engineers and Trigonometry

Modern power companies use alternating current to send electricity over long-distance wires. Electrical engineers use trigonometry to model this flow and the change of direction, with the sine function. Every time you flip on a light switch or turn on the television, you’re benefiting from one of trigonometry's many uses.

5. Manufacturing Industry

Trigonometry plays a major role in this industry, where it allows manufacturers to create everything from automobiles to zigzag scissors. Engineers use trigonometric ratios and trigonometric formulas to determine the sizes and angles of mechanical parts used in machinery, tools, and equipment. This is used in automotive engineering, allowing car companies to size each part correctly and ensure they work safely together. Trigonometry is also used by seamstresses where determining the angle of darts or length of fabric was needed to craft a certain shape of a skirt or shirt is accomplished using basic trigonometric relationships.
 



Summary

  • We realized that trigonometry can be used in many areas such as astronomy and architecture they can aid in calculating many things they can also be used in car, desks and benches. Without really climbing a tree, you can find the height easily with trigonometry. They can be so widely used in real-life applications and is very useful for most architects and astronomers.
  • We can conclude that without trigonometry, life would be much more difficult. Without going through the troubles, you can easily find something so we think that it was a good invention by Archimedes and thanks to this many architects need not go through the trouble to calculate things, so it really helps in real life applications and not only in our tests and exams.

Written by Asha M, Cuemath Teacher 


Frequently Asked Questions (FAQs)

What is sin cos tan?

Sine or Sin theta sin θ = Opposite Side/Hypotenuse           sin θ = 1/cosec θ
Cosine or Cos theta cos θ = Adjacent Side/Hypotenuse cos θ = 1/sec θ
Tangent or Tan theta tan θ = Opposite Side/Adjacent tan θ = 1/cot θ

What are the 3 trigonometric ratios?

The 3 primary trigonometric ratios are sin theta, cos theta and tan theta

How are trigonometric ratios defined?

 

Sine A = Opposite side/Hypotenuse

Cos A = Adjacent side / Hypotenuse

Tan A = Opposite side / Adjacent side

Cot A = Adjacent side / Opposite side

Sec A = Hypotenuse / Adjacent side

Cosec A = Hypotenuse / Opposite side

What is tan?

tan or tangent function is a trigonometric function.

Tangent or Tan theta tan θ = Opposite Side/Adjacent tan θ = 1/cot θ

6 trig functions?

There are 6 trigonometric functions which are:

  • Sine function

  • Cosine function

  • Tan function

  • Sec function

  • Cot function

  • Cosec function

opposite over hypotenuse?

sine fucntion or sin theta is opposite over hypostenuse

 opposite over adjacent?

Tan theta 

 trig ratios?

There are 6 trigonometric functions which are:

  • Sine function

  • Cosine function

  • Tan function

  • Sec function

  • Cot function

  • Cosec function 


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