Tan 60 Degrees

The word ‘Trigonometry’ means measuring the sides of a triangle. An angle is a measure which is the amount of rotation of a revolving line with respect to the fixed line. An angle is positive if the rotation is in anti clockwise direction and if the rotation is in a clockwise direction, then the angle is negative. The two types of conventions for measuring angles are

  • Sexagesimal system
  • Circular system.

In sexagesimal system, consider a unit circle where the unit of measurement is the degree and it is denoted by the symbol ‘10 ‘ . Each 10 is divided into 600 minutes (also denoted as 60’ ) and each minute is subdivided into 60 seconds and it is denoted by 60”.

Trigonometric ratios are represented for the acute angle as the ratio of the sides of a right angle triangle.

Tan 60 Degrees Value

In a right angled triangle, the side opposite the right angle is called the hypotenuse side,the side opposite the angle of interest is called the opposite side and the remaining side is called the adjacent side where it forms a side of both the angle of interest and the right angle.

The tan function of an angle is equal to the length of the opposite side divided by the length of the adjacent side.

\(\tan \theta =\frac{opposite side}{adjacent side}\)

In terms of sine and cosine function, the tangent function is represented by

\(\tan \theta =\frac{\sin \theta }{\cos \theta }\)

Derivation to Find the Value of Tan 60 Degrees

tan 60 degrees

To find the value of tan 60 degrees geometrically, consider an equilateral triangle ABC since each of an angle in an equilateral triangle is 600.

Therefore,\(\angle A=\angle B=\angle C=60^{\circ}\)

Draw a perpendicular line AD from A to BC.

Now consider the triangle, ABD and ADC,

We have, \(\angle ADB=\angle ADC= 90^{\circ}\) and

\(\angle ABD=\angle ACD= 60^{\circ}\)

Therefore, AD=AD

According to AAS Congruency,

\(\Delta ABD \cong \Delta ACD\)

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,\(AD=a\sqrt{3}\)

Now in triangle ADB,

Tan 600= AD/BD

=\(a\sqrt{3}\)/a =\(\sqrt{3}\)

Therefore, tan 60 degrees exact value is given by,

Tan 600=\(\sqrt{3}\)

In the same way, we can derive other values of tan degrees like 00,300,450,600,900,1800,2700 and 3600. Below is the trigonometry table, which defines all the values of tan along with other trigonometric ratios.

trigonometric ratios table

Sample problem:

Question:

Calculate the value tan 90 – tan 270– tan 630 + tan 810

Solution:

Given, tan 90 – tan 270– tan 630 + tan 810

= tan 90 + tan 810 -tan 270– tan 630

= tan 90 + tan (900-90) – tan 270 – tan (900– 270)

= tan 90 + cot 90 -(tan 270+ cot 270) ……(1)

We can write,

tan 90+ cot 90 = 1/ sin 90cos 90

= 2/ sin 180 ……………(2)

tan 270 + cot 270 = 1/ sin 270 cos 270

= 2/ sin 540

= 2/ cos 360 ………..(3)

Substitute (2) and (3) in (1), we get

tan 90 + cot 90 -(tan 270+ cot 270) = (2/ sin 180 )- (2/ cos 360)

= \(\frac{2.4}{\sqrt{5}-1}-\frac{2.4}{\sqrt{5}+1}\)

= 4

Therefore, the value of tan 90 – tan 270– tan 630 + tan 810 = 4

Keep visiting BYJU’S for more information on trigonometric ratios and its related articles, and also watch the videos to clarify the doubts.


Practise This Question

The line x + y = 10 divides line segment AB in the ratio a : 1. Find the value of a.