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.
*Note: This particular blog deals with the following
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
Double- Angle Half Angle
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.
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.
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.
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.
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 functions are defined if f(−x) = −f(x). And the symmetry of the graph is around origin.
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 = c2
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
Cos2 x / Cos2x + Sin2x/ Cos2x = 1 / Cos2x,
(We know from quotient identity that Sin2x/ Cos2x = Tan2x and 1 / Cos2x =Sec2x) 1+ Tan2x = Sec2x (2nd Pythagorean identity)
By dividing first Pythagorean identity equation by Sin2x we get our 3rd equation
Cos2x / Sin2x + Sin2x/ Sin2x = 1 / Sin2x
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
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