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What are Trigonometric Identities?

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09 September 2020                

Read time: 6 minutes

Introduction

Do you find it difficult to understand trigonometry?

Most of us find it difficult to understand Trigonometry as it’s hard to remember so many related formulae and functions. But it can be easy if you understand what is Trigonometry and its functions, how different Trigonometric identities can be proved or derived using the basic relationship of the angles and sides of the triangle.  

Let's first understand what do we understand by Trigonometric Identities.

identities

*Note: This particular blog deals with the following 

  • Reciprocal Identities
  • Pythagorean 
  • Even-Odd

For the rest, please surely visit Trigonometric Identities Part 2

Also read:


What are the basic Trigonometric Identities?

Trigonometric Identities can be defined as trigonometric equations that help us understand and express various relations between the 3 angles and 3 sides of the right-angled triangle.

In short, we can call them Trig identities which are based on Trigonometric functions such as primary functions – Sine, Cosine, and Tangent along with secondary functions – Cosecant, Secant, and Cotangent.

These Trigonometric functions are also defined by different pieces of a Right-Angled Triangle. 

Hence its imperative for us to understand how Trigonometric functions are defined by a right-angled triangle and what Trigonometric identities are based on.

There are many different types of Trigonometric Identities such as

  • Reciprocal Identities
  • Pythagorean
  • Sum-Difference
  • Even-Odd 
  • Double- Angle Half Angle
  • Co-Function
  • Product to Sum
  • Sum to product.  

But now we will discuss only a few important ones from the above list.
Let’s get started with the below diagram of the right-angled triangle which we will refer to in all our explanations.

Trig Identities triangle 1

In the above right triangle O: Opposite side (Height of the Triangle/Perpendicular), A: adjacent side (width/Base), H: hypotenuse (the side opposite to 90°angle)
Once we have the basic right-angle triangle we can easily set up all 6 trig functions.

 

We all know primary trig functions which are Sine, cosine, and tangent, and the way we define these primary Trigonometric functions concerning the above right -angle triangle is based on a mnemonic that we use called SOHCAHTOA.
What it means is in a right triangle 

SOH- Sine of angle ᶱ (Sinᶱ) is equal to the length of the opposite side (O) divided by the length of the Hypotenuse(H) i.e.  Sinᶱ = O/H
CAH- Cosine of angle ᶱ (Cosᶱ) is equal to the length of the Adjacent side (A) divided by the length of the Hypotenuse (H) i.e. Cosᶱ = A/H.
TOA – Tan of angle ᶱ (Tanᶱ) is equal to the opposite side (O) length of the side divided by length of the Adjacent side (A)i.e. Tanᶱ = O/A.
 

If we can remember this SOHCAHTOA concerning the right-angle triangle with these primary trigonometric functions then we can easily build other 3 trigonometric functions and many trigonometric identities.

mnemonic triangle

There is another most commonly used mnemonic to remember the above expressions.
Some People Have 
Sine = Perpendicular/ Hypotenuse
Curly Brown Hair 
Cosine = Base/ Hypotenuse
Turn Permanently Black
Tangent = Perpendicular/Base

From here we have to remember the other 3 trigonometric functions, each of which are the reciprocals of Sine, cosine, and tangent respectively. The above 6 expressions/ trigonometric formulae are the foundation of all trigonometric formulae.
From here we will discuss our first set of trigonometric identities.


Reciprocal Identity

Referring to the above explanation where we discussed Cosec, Sec and cot are reciprocals of Sin, Cos, and Tan the Reciprocal Identities tell us that all these trigonometric functions are somehow reciprocals of each other.
For example 

Sin theta = 1/ Cosec theta   or   Cosec theta = 1/sin theta
Cos theta = 1/Sec theta       
or   Sec theta = 1/Cos theta
Tan theta = 1/Cot theta       
 or   Cot theta = 1/Tan theta

 From the above trigonometric formulae, we can say Cosec is equal to the opposite of sin and reciprocal to each other similarly Cos is equal to the opposite of Sec and reciprocal to each other and Tan is equal to the opposite of Cot and reciprocal to each other.


What are Odd Function/Even Function Identities?

The next one we will discuss is Odd and Even Function identities.

Odd Function

cot fucntion is odd

 Odd functions are defined if f(−x) = −f(x). And the symmetry of the graph is around origin.

Even Function

sec x even fucntion

 Even functions are defined if f(−x) = f(x). And the symmetry of the graph is around y-axis.

Sine function and hence Cosecant function are an odd functions while cosine function and thus, secant function, are even functions. Tan function and Cotan are both odd functions as well. This basically implies:

sin (−x) = −sin (x)

cos (−x) = cos (x)

tan (−x) = −tan (x)

sec (−x) = sec (x)

cosec (−x) = −cosec (x)

cot (−x) = −cot (x)
 


What is Pythagorean Identity?

This one also comes directly from the right-angle triangle. The reason we call them Pythagorean identities is because it is based on the Pythagorean Theorem which is a2 + b2 = c2 . This is also known as the triangle formula.

Here a and b are the length of the 2 legs of the triangle and c is the length of the hypotenuse.
Remember we said Sin theta = a/c or we can say c Sin theta = a. 
We also said Cos theta = b/c or c Cos theta = b. 
Now here if we substitute a & c in Pythagorean theorem with the above trigonometric function, we get

 

a2 + b2 = c
Or (c Cos Θ )2 + (c Sin Θ)2 = c2
c2Cos2Θ + c2Sin2Θ = c2
c2 (Cos2Θ+ Sin2Θ) = c2 
So, after dividing c2/c2 = 1, we get 
(Cos2Θ + Sin2Θ) = 1 

or for any angle x, 

Cos2x + Sin2x = 1 
If we remember how Pythagorean identities are derived directly from the right-angle triangle and also remember SOH CAH TOA which helped us to define a & b that can be plugged to the Pythagorean Theorem. We can easily derive all the trig identities instead of memorizing them!


By diving our first Pythagorean identity by Cos2x we get second equation 
Cosx / Cos2x + Sin2x/ Cos2x = 1 / Cos2x,                                            

(We know from quotient identity that Sin2x/ Cos2x = Tan2x and 1 / Cos2x =Sec2x)
1+ Tan2x = Sec2          (2nd Pythagorean identity)
 

Similarly
 By dividing first Pythagorean identity equation by   Sin2x we get our 3rd equation 
Cos2x / Sin2x + Sin2x/ Sin2x = 1 / Sin2x                                                 

(We know Cos2x / Sin2x = Cot2x and 1 / Sin2x =Cosec2x)
Cot2x + 1 = Cosec2       (3rd Pythagorean identity)
 

Note: The most common Pythagorean identity is the 1st equation and if remember this, we can derive at other 2 Pythagorean identity equation. 

Example 1

What is the value of Cos when Sin  = 5/9 and is positive?

Solution: We know Sin2x + Cos2x = 1
                                    Cos2x = 1- Sin2x
                                    Cos2x = 1-(5/9)2
                                    Cos2x = 1-25/81
                                    Cos2x  = 56/81

Cos x   = √56 / 9


Summary 

In this article, we discussed what trigonometric ratios are briefly and ways to learn them. We covered what trigonometric identities mean, why are they used, and what the different types are.

We also described the first three: Reciprocal Identity, Odd Function/ Even Function Identity, and Pythagoras Formula and Pythagorean Identity in detail with examples. This helped us write the 6 trig functions in an inter-convertible format using the reciprocal identity. We also managed to define the trig ratios as an even function or odd function. Which gave an implication of what sin(-x), cos(-x), tan(-x), cot(-x), sec(-x) and cosec(-x) come out to be.

Lastly, we covered a very important topic of Pythagorean Identities, in which learning the first one can help us derive the other two as well.

Here is the list of them for reference:

Sin theta = 1/ Cosec theta   
Cos theta = 1/Sec theta       
Tan theta = 1/Cot theta        

sn (−x) = −sin (x)

cos (−x) = cos (x)

tan (−x) = −tan (x)

sec (−x) = sec (x)

cosec (−x) = −cosec (x)

cot (−x) = −cot (x)

Cos2x + Sin2x = 1 

1+ Tan2x = Sec2

Cot2x + 1 = Cosec2

There are a few more important identities, their discussion has been continued in Trigonometric Identities Part 2


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FAQs

What are the basic Trigonometric Identities?

  • Reciprocal Identities
  • Pythagorean
  • Sum-Difference
  • Even-Odd 
  • Double- Angle Half Angle
  • Co-Function
  • Product to Sum
  • Sum to product

 What are the basic Pythagorean Identities?

  • Cos2x + Sin2x = 1 
  • 1+ Tan2x = Sec2
  • Cot2x + 1 = Cosec2

External References

 


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