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Sin 30, Cos 30, Tan 30, Sec 30, Cosec 30, Cot 30

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10 September 2020                

Read time: 6 minutes

Introduction

30 degrees or 30 deg is also written as π/6 or pi/6, in radians. For learning about conversion between degrees and radians, please visit Radians and Angles.

Trigonometric Ratios

The most widely used trigonometric Ratios are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, 
So, there are 6 Trig functions, these are :-

  1. Sine                
  2. Cosine            
  3. Tangent
  4. Cosecant
  5. Secant
  6. Cotangent        

Below mention the formulas given in the table for functions of trigonometric ratios(sine, cosine, tangent, cotangent, secant and cosecant) for a right-angled triangle. For more about trigonometric ratios please visit, Trigonometric Ratios 

intro to ratios

sin θ = a/h = Opposite Side/Hypotenuse 
cos θ = b/h =  AdjacentSide/Hypotenuse
tan θ = a/b = Opposite Side/Adjacent
cot θ = b/a =  Adjacent Side/Opposite
sec θ = h/b = Hypotenuse/Adjacent Side
cosec θ = h/a = Hypotenuse/Opposite

Also read:


Sin 30 or sin pi/6

The trigonometric function is also known as a trigonometric ratio or an angle function is used to relate the angles of a triangle to its sides.

Sin 30 Value

The value of the sin 30 degrees or sin pi/6 is 0.5 or 1/2

For angles less than a right angle, trigonometric functions are commonly defined as the ratio of two sides of a right triangle. Here, we will discuss the value for sin 30 degrees and we were taught how to derive the sin 30 value using other degrees or radians.

                          Sinθ = Opposite side/Adjacent Side

Geometrically Derivation of sine 30

The derivative of f(x) = sin x is given by
f '(x) = cos x
Now, calculate the sin 30 value. Consider an equilateral triangle ABC. Considering each angle in an equilateral triangle is 60°, therefore 
                                             ∠A=∠B=∠C=60°

sin 30 derivation

Draw the perpendicular line AD from A to the side BC (From above figure)
Now,        ΔABD≅ΔACD

Therefore BD=DC and also
∠BAD=∠CAD

Now observe that the triangle ABD is a right triangle, with angle D as 90 degrees with 
∠BAD = 30 degrees and ∠ABD = 60 degrees
And you know very well, for finding the trigonometric ratios, we need to know some things about the lengths of the sides of the triangle.

So, let us suppose that AB=2a, BD=1/2, BC=a
 
To find the sin 30-degree value, and use sin 30 degree formula we use:
Sin 30° = opposite side/hypotenuse side

 

We know, Sin 30° = BD/AB = a/2a = 1 / 2
Accordingly, Sin 30 degree equals to the fractional value of 1/ 2.

Sin 30° = 1 / 2

or sin pi/6 = 1/2
 

Therefore, sin 30 value is ½

Why is sin 30 is equal to sin 150?

The value of sin 30 degrees = sin 150 degrees 
Sin 30 = sin 150 = ½

sin 150 equal to sin30

Both are equal because the reference angle for 150 is equal to 30 for the triangle formed in the unit circle. The reference angle is formed when the perpendicular is dropped from the unit circle to the x-axis, which forms a right triangle.
Since the angle 150 degrees lies on the IInd quadrant, therefore the value of sin 150 is positive. The internal angle of the triangle is 180 – 150=30, which is the reference angle.

Sin(-30)

sin(-x) = -sin(x)  (odd function)

Hence, sin (-30) = -sin 30 

sin(-30) = -1/2


Cos 30 or cos pi/6

In trigonometry, the cosine function is defined as the ratio of the adjacent side to the hypotenuse

Cos 30 value

The value of Cos 30 degree is √3/2
Another Different form of Cos 30° is cos pi/6 or cos π/6
Hence, cos pi/6 =  √3/2

 

Cos 30° = √3/2 is an irrational number and it equals 0.8660254037 approx.
And then, the exact value of cos 30 degrees is written as 0.8660 approx.
√3/2 is the value of Cos 30° which is a trigonometric function of a particular angle.
                          
Law of Cosine
In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. According to the law,
                a2 = b2 + c2 − 2bc. cos (A)
                b2 = a2 + c2 − 2ac. Cos (B)
                c2 = a2 + b2 − 2ab. Cos (C)              

Geometrically Derivation of cosine 30

The derivative of f(x) = cos x is giving below
f '(x) = - sin x
Now we calculate the value of Cos 30 degrees through a practical approach. So if we drew a right triangle using constructions, Cos pi/6 can be calculated as √3/2

                        cos 30 derivation

Cos 150?

Since cos (180-x) = -cos x,

cos 150 = - cos 30

cos 150 = cos 5pi/6 = -√3/2

Cos(-30)

cos(-x) = cos(x)  (even function)

Hence, cos (-30) = cos 30 

cos(-30) = √3/2


Tan 30 or tan pi/6

In trigonometry, the tangent of an angle in a right-angled triangle is equivalent to the ratio of the opposite side and the adjacent side.

Tan 30 value

The value of tan 30 degrees is 1/√3
Tan 30 degrees is also expressed by tan π/6 in terms of radians. 

Hence, tan pi/6 is also 1/√3

The exact value of tan 30° is 0.57735.
The value of the tangent of angle 30 degrees can also be evaluated using the values of sin 30 degree and sin 30 degree.      

Geometrically Derivation of Tangent 30

The derivative of f(x) = tan x is given by
f '(x) = sec 2 x
Now we can find tan 30 Value Using Sin and Cos Function

tan 30 derivation                              

We can also represent the tangent function as the ratio of the sine function and cosine function.
∴ Tan a = sin a/cos a
So, tan 30 degrees we can expressed as ;

Tan 30° = sin 30°/cos 30°
We know,
Sin 30° = 1/2  &  Cos 30° = √3/2
∴ Tan 30° = (1/2) /(√3/2)

 

Tan 30° = 1/√3
So, the value of Tan 30 degrees or tan pi/6 is 1/√3

Tan(-30)

tan(-x) = -tan(x)  (odd function)

Hence, tan(-30) = -tan30 

tan(-30) = -1/√3


Cot 30 or cot pi/6

In trigonometry, the Cotangent function is defined as the ratio of the adjacent side to the Opposite side.

Cot 30 value

The value of the Cot 30 degree or cot pi/6 is √3

                           
Derivation of Cot pi/6

The derivative of f(x) = cot x is given by
f '(x) = - csc 2 x

cot 30 derivation

cot  x = 1/tan x.

Using this relation, we can find cot pi/6

cot pi/6 = 1/tan pi/6

Therefore, cot 30 or cot pi/6 is √3

Cot(-30)

cot(-x) = -cot(x)  (odd function)

Hence, cot(-30) = -cot30 

cot(-30) = -√3


Cosec pi/6 or cosec 30 

In trigonometry, the cosecant of an angle is the length of the hypotenuse divided by the length of the opposite side.
it is abbreviated to just 'csc'.

Value of Cosec 30 or Cosec pi/6

The value of cosec 30 degrees or cosec pi/6 is 2.

It can be derived as the reciprocal of sine:
Cosec = 1/Sin
The exact value of cosec 30° is −1.012113353                     

Geometrically Derivation of Cosecant 30                                            

The derivative of f(x) = csc x is given by
f '(x) = - csc x cot x
Take a triangle ABC with each side equal to 2a 

cosec 30 derivation


Angle A = B = C

Draw AD perpendicular to BC

ΔABD ≅  ΔACD by RHS,

BD= CD and angle BAD = angle CAD - 30 degrees each

by Pythagoras theorem, AD = √3a

Take ΔABD, cosec 30 = AB/BD = 2a/a = 2

Hence, the value of cosec 30 is 2.


Secant 30 degree

In Trigonometry, the secant function defines a relation between the hypotenuse and adjacent side of the right-angled triangle.

Value of sec 30 or sec pi/6

The value of Sec 30 degrees is 2/√3.
 

sec 30 derivation

Secant function is the transpose of the cosine function                                                   
Sec ∠α = Hypotenuse/Adjacent Side
             = Hypotenuse/Base
sec ∠α = h/b

 

Now, sec ∠α = 1/cos ∠α
Therefore, sec 30 = 1/cos 30


Geometrically Derivation of Secant 30

sec 30 = 1/cos 30

sec 30 derivation

Therefore, 

sec 30 or sec pi/6 is 2/√3. 


Summary

In Trigonometry, measurement, and calculation of the lengths of sides and the sizes of angles of triangles is dealt with. The most widely used trigonometric functions are the sine, the cosine, and the tangent and the remaining three functions we use rarely which are secant, cosecant, and cotangent =. 30 deg can also be used as pi/6 and all the trigonometric values of trig functions can be derived for the angle. 

 𝚹 (°)

   30°

 𝚹(rad)

   π/6

 Sine

   1/2

 Cosine        

   √3/2

 Tan

   1/√3

 Cot

   √3

 Cosec

   2

 Sec

   2/√3

Written by Neha Tyagi, Cuemath Teacher


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