Learn US

Sin pi/3, Cos pi/3, Tan pi/3, Sec pi/3, Cosec pi/3, Cot pi/3

976 views

11 September 2020                

Read time: 7 minutes

Introduction

60 degrees or 60 deg is also written as π/3 or pi/3, in radians. For learning about conversion between degrees and radians, please visit Radians and Angles.

Trigonometry is the study of the relationship between the sides and angles of the triangle. The trigonometric ratios have their names and abbreviations:- sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). These terms can be determined from the ratios of the sides of the triangle. In this blog, we will study the trigonometric ratios of 60°.

60 degrees triangle real example

Do you wonder how can we find the length of the ladder in this situation? At the end of the article, you will be in a position to find it!
It has been noticed that students cannot actually remember values of six trigonometric ratios (sin, cos, tan, cosec, sec, and cot) for 0°, 30° 45° 60°, and 90° These values are used very often and it is recommended that student should be able to tell the values instantly whenever asked.
Let’s study it with the following diagram.
Explanation:
Trigonometric ratios of 60 degrees (and even 30 degrees ) are based on an equilateral triangle with sides of length 2 units each and with one of the angles bisected. We get two congruent triangles (exactly the same size). The three angles of each of these half triangles are 30°, 60°, and 90° as shown in the following figure.

60 deg triangle   

Draw the perpendicular line RS from R to the side PQ (From above figure)
Now,        ΔRPS ≅ ΔRQS
Therefore, SP = SQ and also
∠PRS = ∠QRS
Now observe that the triangle RSP is a right triangle, with angle S as 90 degrees with 
∠PRS = 30 degrees and ∠RPS = 60 degrees
 
You have already learnt the trigonometric values of all ratios.  Remember that All trigonometric functions can be written as sohcahtoa,  ”SOH” – Sine is Opposite over Hypotenuse, "CAH" - meaning Cosine is Adjacent over Hypotenuse. ”TOA” – Tangent is Opposite over Adjacent, and their reciprocals can also be deduced in the same manner for cosec x, sec x, and cot x.

For more about trigonometric ratios please visit, Trigonometric Ratios 


Sin 60 or sin pi/3

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 60 Value

The value of the sin 60 degrees or sin pi/3 is √3/2
Sin 60° = √3/2 is an irrational number and it equals 0.8660254037 approx.
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 60 degrees and we were taught how to derive the sin 60 value using other degrees or radians.
                          Sinθ = Opposite side/Adjacent Side

Geometrically Derivation of sine 60

In ΔRPS, 
we get for the values of 60, the opposite is √3 units, the hypotenuse is 2 units and adjacent is 1 unit by Pythagoras theorem.

sin 60   =    opposite/hypotenuse = √3/2
 
so,  sin 60 = √3/2

Why is sin 60 is equal to sin 120?

The value of sin 60 degrees = sin 120 degrees 
Sin 60 = sin 120 = √3/2

sin 60 equal to sin 120

Both are equal because the reference angle for 120 is equal to 60 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 120 degrees lies on the IInd quadrant, therefore the value of sin 120 is positive. The internal angle of the triangle is 180 – 120 = 60, which is the reference angle.

Sin(-60)

sin(-x) = -sin(x)  (odd function)
Hence, sin (-60) = -sin 60 
sin(-60) = -√3/2


Cos 60 or cos pi/3

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

Cos 60 value

The value of Cos 60 degree is 1/2
Another Different form of Cos 60° is cos pi/3 or cos π/3
Hence, cos pi/3 =  1/2

Geometrically Derivation of cosine 60

Now we calculate the value of Cos 60 degrees through a practical approach. So if we drew the right triangle RPS using constructions, 
Cos pi/3 can be calculated as ½
OR
Using Pythagorean identity, we get
 
sin260°+ cos260°=1
cos260°=1 -sin260°  
cos260°=1- (√3/2)2
cos260°=  1- 3/4=  14 
Hence, cos 60 =  1/2.

Cos 120?

Since cos (180-x) = -cos x,
cos 120 = - cos 60
cos 120 = cos 2pi/3 = -1/2

Cos(-60)

cos(-x) = cos(x)  (even function)

Hence, cos (-60) = cos 60 

cos(-60) = 1/2


Tan 60 or tan pi/3

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 60 value

The value of tan 60 degrees is √3
Tan 60 degrees is also expressed by tan π/3 in terms of radians. 
Hence, tan pi/3 is also √3
The approx value of tan 60° is 1.732
The value of the tangent of angle 60 degrees can also be evaluated using the values of sin 60 degrees and cos 60 degrees.    

       

Geometrically Derivation of Tangent 60

Now we can find tan 60 Value Using the Sine function 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 60 degrees we can expressed as ;
Tan 60° = sin 60°/cos 60°
We know,
Sin 60° = √3/2 &  Cos 60° = ½
 ∴ Tan 60° = (√3/2)/(1/2) 

Tan 60° = √3
So, the value of Tan 60 degrees or tan pi/3 is 1/√3

Tan(-60)

tan(-x) = -tan(x)  (odd function)
Hence, tan(-60) = -tan60 
tan(-60) = -√3


Cot 60 or cot pi/3

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

Cot 60 value

The value of the Cot 60 degree or cot pi/3 is 1/√3

Derivation of Cot pi/3

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/3

cot pi/3 = 1/tan pi/3

Therefore, cot 60 or cot pi/3 is 1/√3

Cot(-60)

cot(-x) = -cot(x)  (odd function)

Hence, cot(-60) = -cot60 

cot(-60) = -1/√3


Cosec pi/3 or cosec 60 

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 60 or Cosec pi/3

The value of cosec 60 degrees or cosec pi/3 is 2/√3.
It can be derived as the reciprocal of sine:
Cosec = 1/Sine
The approx value of cosec 60° is 1.1547                  

Geometrically Derivation of Cosecant 60                                            

cosec 60 = 1/sin 60 
Hence, the value of cosec 60 is 2/√3..


Sec 60 or sec pi/3

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

Value of sec 60 or sec pi/3

The value of Sec 60 degrees is 2
 

sec 30 derivation

Secant function is the transpose of the cosine function                                                   

Geometrically Derivation of Secant 60

Sec x = Hypotenuse/Adjacent Side
             = Hypotenuse/Base
sec x = h/b

Now, sec ∠A = 1/cos ∠A
Therefore, sec 60 = 1/cos 60
sec 60 = 1/cos 60
Therefore, 
sec 60 or sec pi/3 is 2 


There is yet another way of finding out the values of all trigonometric ratios. For learning the Trigonometric values table much quicker please visit: Trigonometry Table 

20 60 90 ladder

Remember that you had to calculate the length of the above ladder….
So we can see the triangle is right-angled and the hypotenuse is the ladder. Using the height of the wall and given angle, we can calculate the length of the ladder as follows:-
hypotenuse/perpendicular= cosec 60°
hypotenuse=6 × 2/√3 = 6.94 m.
So the length of the ladder is 6.94m.


Complementary Terms

Did you notice how we reverse the values of sine to get the values of cosine in the Trigonometry table? Because they are complementary of each other.

What are complementary angles?

When sum of two angles is 90 then they are called complementary angles.

complementary angles

For example,
How do we prove that the sine of an angle equals the cosine of its complementary angle?
Consider the following right triangle ABC. When we find the sine and cosine of angles A and B in terms of lengths of the sides a, b, and c, what do we notice?

complementary angle triangle

sinA = a/c,            sinB = b/c,
cosA = b/c,            cosB = a/c
 
therefore,  sinA = cos B and sin B = cos A.
Since the triangle is a right-angled triangle so the sum of A and B will always be 90 degrees.
That means A and B are complementary to each other. For complementary angles, all terms starting with “co…” will be complementary of the terms without “co”.
You will see that

sin(θ) = cos(90 –θ)
cos(θ) = sin(90 −θ)
tan(θ) = cot(90 −θ)
cot(θ) = tan(90 −θ)
sec(θ) = cosec(90 −θ)
cosec(θ) = sec(90 −θ)

30 60 complementary

so we see that sin 30 degrees = cos 60 degrees,  
Similarly, cos 30 degrees =  sin 60 degrees

 


Summary

I hope this article has helped you to understand trigonometric angles of 60 deg or  3 radians. In higher classes, you will learn that angles may vary as any real value. And if we know the value of basic angles then many other values can be calculated in a fraction of seconds.

𝚹 (°)

  60°

𝚹(rad)

  π/3

Sine

  √3/2

Cosine        

1 /2

Tan

  √3

Cot

  1/√3

Cosec

  2/√3

Sec

  2

 

Written by Anupama Mahajan, Cuemath Teacher


GIVE YOUR CHILD THE CUEMATH EDGE
Access Personalised Math learning through interactive worksheets, gamified concepts and grade-wise courses
Learn More About Cuemath