Tan 30 Degrees: Value of tan 30 with Proof, Examples and FAQ

Tan 30 degrees

The value of the tangent of the angle 30° in a right-triangle is called tan of angle 30 degrees. The tangent of angle 30° is a value representing the ratio of the opposite side’s length to the adjacent side’s length with respect to 30° angle.

In trigonometry, we write tan (30°) mathematically, and its exact value in fraction form is 1/√3. Therefore, we write it in the following form in trigonometry.

tan (30°) = tan π/6 = 1/√3

Value of Tan 30°

The exact value of tan π/6 is 1/√3 equal to 0.5773502691… in decimal form. It is reciprocal of cot 30 degrees. The approximate value of the tangent of angle 30° is equal to 0.57735.

tan (30°) = 0.5773502691… ≈ 0.57735

Proof

The exact value of tan π/6 can be derived using three methods explained below.

• Theoretical Method

The exact value of tan (30°) can be derived on the basis of geometrical relations between sides of the right-angled triangle when one of the angles of the triangle is 30 degrees.

According to the properties of a right-angled triangle, if one of the angles of the right triangle is 30°, then the length of the adjacent side to 30° is √3/2 times the length of the hypotenuse.

Therefore, in △OPQ, 

OQ = (√3/2) × OP

or OP = (2/√3) × OQ

Now, apply the Pythagorean Theorem: 

\text { Hypotenuse }^{2}=\text { Perpendicular }^{2}+\text { Adjacent } \text { Side }^{2}

\mathrm{OP}^{2}=\mathrm{OQ}^{2}+\mathrm{PQ}^{2}

Substituting the value of OP in terms of OQ

\Rightarrow(2 / \sqrt{3})^{2} \times \mathrm{OQ}^{2}=\mathrm{OQ}^{2}+\mathrm{PQ}^{2}
\Rightarrow(4 / 3) \mathrm{OQ}^{2}=\mathrm{OQ}^{2}+\mathrm{PQ}^{2}
\Rightarrow(4 / 3) \mathrm{OQ}^{2}-\mathrm{OQ}^{2}=\mathrm{PQ}^{2}
\Rightarrow(1 / 3) \mathrm{OQ}^{2}=\mathrm{PQ}^{2}
\Rightarrow \mathrm{PQ}^{2} / \mathrm{OQ}^{2}=1 / 3
\Rightarrow \mathrm{PQ} / \mathrm{OQ}=1 / \sqrt{3}

PQ and OQ are lengths of opposite and adjacent sides of the right-angled triangle.

⇒ Length of opposite side/adjacent side = 1/√3

The angle of △OPQ is 30°.

Therefore, we can write that tan (30°) = 1/√3.

• Practical Method

You can also find the value of the tangent of angle 30° practically by constructing a right-angled triangle with a 30° angle by geometrical tools.

Draw a straight horizontal line from Point H and then construct an angle of 30° using the protractor.

Set compass to any length by a ruler. Here, the compass is set to 7 cm. Now, draw an arc on the 30° 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 30 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 3.5 cm, and the length of the adjacent side is 6.05 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 30°.

tan (30°) = IJ/GJ = (3.5)/(6.05)

So, tan (30°) = 0.578512396… ≈ 0.57735

• Trigonometric Method

We can prove the value of tan (30°) with a trigonometric approach.

we know that sin 30° = (½) and cos 30° = (√3/2)

Also, by trigonometric identities,

sin x/cos x = tan x

Put x = 30°

tan (30°) = sin (30°)/cos (30°)

Substitute the values of sin 30° and cos 30°

tan (30°) = (½)/(√3/2)

tan (30°) = 1/√3

Hence, we proved the value of tan (30°) using different approaches.

Example

1. Evaluate: tan 30° + sin 30°

Solution:

We know that tan (30°) = 1/√3 and sin (30°) = 1/2
So, tan (30°) + sin (30°)
= 1/√3 + 1/2
= (2+√3)/2√3

2. Evaluate: 2 tan 30° – 2 cos 30°

Solution:

We know that tan (30°) = 1/√3 and cos (30°) = √3/2
So, 2 tan (30°) – 2 cos (30°)
= 2 (1/√3) – 2(√3/2)
= 2/√3 – √3
= -(1/√3)