Tan 60 Degrees: Value of tan 60 with Proof, Examples and FAQ

Tan 60 degrees

The value of the tangent of the 60°angle in a right-angled triangle is called tan of angle 60 degrees. The tangent of angle 60° is a value representing the ratio of the opposite side’s length to the adjacent side’s length with respect to the considered angle.

In trigonometry, we write tan (60°) mathematically, and its exact value in fraction form is √3. Therefore, we write it in the following form in trigonometry:

tan (60°) = tan π/3 = √3

Value of Tan 60°

The exact value of tan(π/3) is 1/√3 equal to 1.7320508075… in decimal form.

It is reciprocal of cot 60 degrees.

The approximate value of the tangent of angle 60 degrees is equal to 1.7321.

tan (60°) = 1.7320508075… ≈ 1.7321

Proof

The exact value of tan (π/3) can be derived using three methods explained below.

• Theoretical Method

We can derive the exact value of tan (60°) by considering an equilateral triangle ABC.

We know that each angle in an equilateral triangle is 60°.

So, ∠A = ∠B = ∠C = 60°

Now, draw a perpendicular line AD from point A to side BC.

Now we have two right-angled triangles ABD and ADC.

Here, ∠ ADB = ∠ADC = 90° and,

∠ ABD = ∠ACD = 60°

AD = AD

According to AAS Congruency

Δ ABD ≅ Δ ACD

From this, we conclude that

BD = DC    (since they are corresponding parts of congruent triangles)

Take the value of AB = BC = 2a

Then, BD =  ½ (BC)

 = ½ (2a) = a

Now use Pythagoras theorem in the △ABD

\mathrm{AB}^{2}=\mathrm{AD}^{2}-\mathrm{BD}^{2}
\mathrm{AD}^{2}=\mathrm{AB}^{2}-\mathrm{BD}^{2}
\mathrm{AD}^{2}=(2 \mathrm{a})^{2}-\mathrm{a}^{2}
\mathrm{AD}^{2}=4 \mathrm{a}^{2}-\mathrm{a}^{2}
\mathrm{AD}^{2}=3 \mathrm{a}^{2}
\text { So, } \mathrm{AD}=\mathrm{a} \sqrt{3}

Now in right-angled triangle ADB,

tan (60°) = (opposite side to the ∠ ABD) / (adjacent side to the ∠ ABD)

tan (60°) = AD/BD

tan (60°) = a√3/a 

                = √3

Therefore, tan (60°) =√3

• Practical Method
  

You can also find the value of the tangent of angle 60° practically by constructing a right-angled triangle with a 60° angle by geometrical tools.

Draw a straight horizontal line from Point H and then construct an angle of 60° using the protractor.

Set the compass to any length by a ruler. Here, the compass is set to 4.4 cm. Now, draw an arc on the 60° angle line from point H, and it intersects the line at point I.

Finally, draw a perpendicular line on the horizontal line from point I, and it intersects the horizontal line at point J perpendicularly. Thus, a right-angled triangle ∆HIJ is formed.

Now,  calculate the value of the tangent of 60 degrees and for this, measure the length of the adjacent side (HJ) with a ruler. You will observe that the length of the opposite side (IJ) is 3.8 cm, and the length of the adjacent side (HJ) is 2.2 cm in this example.

Now, find the ratio of lengths of the opposite side to the adjacent side and get the value of the tangent of angle 60°.

Here,

tan (60°) = (opposite side) / (adjacent side)

tan (60°) = IJ/HJ = (3.8)/(2.2)

So, tan (60°) = 1.727272… ≈ 1.7321

• Trigonometric approach

We can prove the value of tan (60°) with a trigonometric approach.

we know,

sin 60° = √3/2,

cos 60° = 1/2

Also, by trigonometric identities,

sin x/cos x = tan x

Put x = 60°

tan (60°) = sin (60°)/cos (60°)

Put the values of sin 60° and cos 60°

tan (60°) = (√3/2)/(1/2)

tan (60°) = √3

Hence, we proved the value of tan (60°) using different approaches.

Example

1. Evaluate: tan 60° + sin 60°

Solution:

We know that tan (60°) = √3 and sin (60°) = √3/2
So, tan (60°) + sin (60°)
= √3 + √3/2
= (3√3)/2

2. Evaluate 2 tan 60° – 2 cos 30°

Solution:
We know that tan (60°) = √3 and cos (30°) = √3/2
So, 2 tan (60°) – 2 cos (30°)
= 2√3 – 2(√3/2)
= 2√3 – √3
= √3