Tan 60 Degrees

Value Of Tan 60 Degree

Trigonometry is a branch of Mathematics that deals with triangles. The word Trigonometry is composed of two Greek words trigōnon (meaning triangle) and metron (meaning measure).  So in short, we can say that measuring a triangle (specifically right-angled triangle) is trigonometry. Trigonometry is the study between the relationships dealing with angles, heights and lengths of triangles and also the relationships between the different circle parts and other geometric figures. In the field of astronomy, engineering, architectural design, and physics, trigonometry applications are found. Trigonometric identities are very useful and help to solve the problems better by studying the formulae below. There are a huge number of fields in which these trigonometry identities and trigonometric equations are used. 

Tan 60 Degrees Value

A right-angled triangle is a closed figure having three sides, three angles and three edges such that one of the three angles of a triangle is of 90degree.

                                

To find the value of tan 60 degrees, let us take one of the angles 60 degrees. 


  • Hypotenuse: The longest side of all the three sides of a right-angled triangle which is also opposite to the right angle is called Hypotenuse.

  • Base or Adjacent: The side containing 60 degrees as well as 90 degrees is called Base.

  • Perpendicular or Opposite: The side perpendicular to base not containing 60 degrees is called perpendicular or opposite side of a right-angled triangle.


\[\sin \left( \theta  \right) = \frac{{opposite}}{{Hypotenuse}}\]  

 \[\cos \left( \theta  \right) = \frac{{Adjacent}}{{Hypotenuse}}\]

Here, = 60 degrees.


\[\tan \left( \theta  \right)\frac{{\sin \theta }}{{\cos \theta }}\]

Thus, 


\[\frac{{\sin \left( \theta  \right)}}{{\cos \left( \theta  \right)}} = \frac{{\frac{{opposite}}{{Hypotenuse}}}}{{\frac{{Adjacent}}{{Hypotenuse}}}} = \frac{{Opposite}}{{Adjacent}} = \tan \left( \theta  \right)\]

\[\tan \left( \theta  \right) = \frac{{Opposite}}{{Adjacent}}\]


Therefore, the value of tan 60 = Opposite / Adjacent.

What is the value of tan 60 Degrees?                      

We can use geometry to find the value of tan 60 degrees. Let us consider an equilateral triangle ABC such that all the interior angles are 60 degrees.

Therefore, \[\angle A = \angle B = \angle C = 60^\circ \]


A perpendicular AD is drawn from A to BC. In triangle ADB we can say \[\angle {\text{ }}ADB = 90^\circ \]and in triangle ADC,  \[\angle ADC = 90^\circ .\]Thus,  ∠ ADB =  ∠ ADC. 

We also know that

\[\angle {\text{ }}ABD = \angle ACD = {\text{ }}60^\circ \]

That means, AD=AD


Now, the question arises what is the value of the tangent of 60 degrees. So, we know that according to AAS Congruency,

\[\Delta {\text{ }}ABD{\text{ }} \cong {\text{ }}\Delta {\text{ }}ACD\]

From this, we can say

BD = DC

Let us take, AB = BC =2a

Then, 

BD= ½ (BC) =½ (2a) =a

By using Pythagoras theorem,

  • AB2 = AD2- BD2

  • AD2 = AB2- BD2

  • AD2 = (2a)2- a2

  • AD2 = 4a2- a2

  • AD2 = 3a2

Therefore, AD=a√3

Now in triangle ADB,

Tan 60 = AD/BD = a√3/a = √3

Therefore, the exact value of Tan 60 degrees is √3. We can also derive the values of tan 0°, 30°, 45°, 90°, 180°, 270° and 360° in the same way. 


The values of the important functions of Sin can also be determined by the given method:


Simplifying in a tabular form:


30°

45°

60°

90°

Sin

0

\[\frac{1}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{{\sqrt 3 }}{2}\]

1


The value of cosine functions is opposite if sine functions as in:


Summary table of the value of sin and cosine angles:


30°

45°

60°

90°

Sin

0

\[\frac{1}{2}\]

\[\frac{{\sqrt 1 }}{{\sqrt 2 }}\]

\[\frac{{\sqrt 3 }}{2}\]

1

Cos

1

\[\frac{{\sqrt 3 }}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{1}{2}\]

0


From here, we can find out the value of tan for each angle by:

                                           \[\tan \left( \theta  \right)\frac{{\sin \theta }}{{\cos \theta }}\]

Summary of the values of all the trigonometric functions:

Trigonometry Table

Angle

    0o

    30o

    45o

    60o

  90o

Sin \[\theta \]

    0

\[\frac{1}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{{\sqrt 3 }}{2}\]

  1

Cos \[\theta \]

    1

\[\frac{{\sqrt 3 }}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{1}{2}\]

  0

Tan \[\theta \]

    0

\[\frac{1}{{\sqrt 3 }}\]

      1

\[\sqrt 3 \]

  \[\infty \]

Cosec \[\theta \]

    \[\infty \]

  2 

\[\sqrt 2 \]

\[\frac{2}{{\sqrt 3 }}\]

  1

Sec \[\theta \]

    1

  \[\frac{2}{{\sqrt 3 }}\]

\[\sqrt 2 \]

      

    \[\infty \]

Cot \[\theta \]

  \[\infty \]

  

  \[\sqrt 3 \]

1

\[\frac{1}{{\sqrt 3 }}\]


0



Solved Examples:

Question: Calculate the value of Tan 75 degrees.

Solution: The value of tan 75 degrees can be calculated in many ways. One of the ways is given below:


75 can also be seen as a sum of 30 and 45 (because the value of tan 30 nd tan 45 can easily be placed as per the table)

 tan75 = tan(30+45)

Also, 

tan(A+B) = (tanA + tanB)/(1 - tanAtanB)

So, tan(30+45) = (tan30+tan45)/(1+tan30tan45)

Now, tan30 = 1/✓3

And, tan45 = 1

tan(30+45) = (1/✓3 + 1)/(1+ 1/✓3)

Therefore, tan75 = (1 + 1/✓3)²

So, tan75 = 1 + 1/3 + 2/✓3 

        tan75 = 4/3 + 2/✓3

Thus, tan75 = (4✓3 + 6)/3✓3

The value of✓3 is 1.73205

So, tan75 = 12.9282/5.19615 = 2.4880344

So, the approximate value of tan75 is 2.4880344


FAQ (Frequently Asked Questions)

Q1: What is Pythagoras Theorem?

Ans: Pythagoras theorem is applicable only on a right-angled triangle. It explains that in a right-angle triangle, the square of the hypotenuse will always be equal to the sum of the squares of the opposite side and the square of the adjacent side.

Where, c = hypotenuse, a = base and b = perpendicular. 

Pythagoras Theorem formula;

(Hypotenuse)2= (Base)2+ (Perpendicular)2

Q2: What are the Trigonometric Identities?

Ans: Trigonometric identities are the equations that follow Pythagoras theorem which holds true for a right-angled triangle.