Tan 30 Degree

There are three primary trigonometric ratios in trigonometry, which are sine, cosine, and tangent. All the trigonometric formulas, trigonometric functions, and identities are based on these primary ratios. These trigonometric ratios are very useful to solve angles and sides in a right-angled triangle. Sin, cos, and tan are the ratios that are used to calculate two angles, where the degrees like 0°, 30°, 45°, 60°, and 90° are very common in solving the problem. Tan 30° has equal importance, like all other trigonometric ratios. 

What is the Value of Tan 30 Degrees?

In a right-angled triangle, the ratio of the perpendicular side to the adjacent side is equal to the tangent of the angle. Hence, to find the value of the tan 30 degrees, the hypotenuse must be slanted at 30° degree with the base, and the value of the perpendicular (opposite) side and the adjacent (base) side of the right-angled triangle must be known. 

Here are the different values of tan 30°. We will also discuss the methods required to find the value of tan 30 degrees with examples.

• Tan 30°: 1/√3

• Tan 30° in decimal: 0.5773502. . .

• Tan (-30 degrees): -0.5773502. . . or -1/√3

• Tan 30° in radians: i.e, tan (π/6) or tan (0.5235987 . . .)

We can also represent the tan 30 degrees as:

• sin(30°)/cos(30°)

• ± sin 30°/√(1 - sin²(30°))

• ± √(1 - cos²(30°))/cos 30°

• ± 1/√(cosec²(30°) - 1)

• ± √(sec²(30°) - 1)

• 1/cot 30°

Note that, since 30° lies in the 1st quadrant, the final value of tan 30° will also be positive, and it is an irrational number.

We can use trigonometric identities to represent tan 30° as,

• cot(90° - 30°) = cot 60°

• -cot(90° + 30°) = -cot 120°

• -tan (180° - 30°) = -tan 150°

Methods to Find Value of Tan 30 Degrees

The tangent function is always positive in the 1st quadrant. The value of tan 30° is also given as 0.57735. . .. We can find the value of tan 30 degrees by:

Using Trigonometric Functions:

We can find tan 30° using sin and cos. The values of the sin 30° and cos 30° are used to find the value tan of 30°, but the condition is that sin 30°, and cos 30° must be from the same triangle. It is just a very basic concept of trigonometry to find the tangent of the angle using the sine and cosine of the angle. It is known that the ratio of sine and cosine of the same angle gives the tangent of the same angle. So, if we have the value of sin 30° degree and cos 30° degree, then the value of tan 30° degrees can be calculated very easily using the above given trigonometric relations.

Using Unit Circle: 

We can calculate the value of tan 30 degrees from a unit circle. The unit circle features a radius of 1 unit, and it is drawn on an XY plane. With the below graph, you will check the values of all the trigonometry ratios, like sin, cos, tan, sec, cot, and cosec.

• Rotate ‘r’ anticlockwise. It has to form a 30° angle with the positive x-axis.

• The tan of 30 degrees equals the y-coordinate (0.5) divided by the x-coordinate (0.866) of the point of intersection (0.866, 0.5) of the unit circle and r.

• Hence the value of tan 30° is y/x = 0.5774 (approx).

Finding Tan 30° Degrees Value

Like sine and cosine, the tangent is additionally a basic function of trigonometry. Most of the trigonometric equations are based on these ratios. Usually, to find the values of sine, cosine, and tangent ratios, we use right-angled triangles and also take a unit circle example. First, we will discuss the tan 30° value in terms of a right triangle.

Suppose for a triangle ABC, right-angled at C, α is the angle, h is the hypotenuse, b is the adjacent side or base, and a is the opposite side or perpendicular. As we know,

Tan α =Opposite Side/Adjacent Side

∴ Tan α=a/b

Similarly, we can also find the value for Sine and Cosine ratios.

Sin α= Opposite Side/Hypotenuse

∴ Sin α =a/h

And,

Cos α = [Adjacent Side/Hypotenuse

∴ Cos α = b/h

We can also represent the tangent function as the ratio of the sine function and cosine function as:

∴ Tan α = sin α/cos α

So, we can write tan 30 degrees as;

Tan 30° = sin 30°/cos 30°

We know,

Sin 30° = 1/2 

Cos 30° =√3/2

∴ Tan 30° = (1/2) /(√3/2)

Tan 30° = 1/√3

Hence it is proved that the value of Tan 30° is 1/√3

Practical Uses of Trigonometry

• In our daily lives, the trigonometry ratios of sin, cos, and tan are used for calculating heights and distances. Many real-life problems are also solved using sin, cos, and tan. 

• Trigonometry is also used to calculate the planetary motions and total distances.

• It is used to calculate the angle of elevation, slope and heights of buildings and mountains, etc.

• It is used in the structural field to calculate the loads, roof slopes, flooring, etc.

• Trigonometry is used in various flight engineering to calculate the angle change due to wind loads used in aviation.

• It is even used by Archeologists to calculate the excavations work.

• It is used in satellite systems.

• Trigonometry is used in marine engineering to design various marine ramps.

Practice Questions 

1. Two poles of equal heightare standing to each other on either side of a road, which is 30 m wide. To some extent between them on the road, the angles of elevation of the tops are 30° and 60°. The height of each pole is?

Ans: Let AB and CD be the poles and let O be the point of observation.

Let AB = CD = h m

∠AOB = 30°, ∠COD = 60°.

Let OA = x.

Then, OC = (30 - x) m

In △BAO, x/h = cot30° = √3

= x = h√3 ….(1)

In △DCO, 30 - x/ h = cot60° = 1/√3

= 30 - x = h/√3

= x = (30 - h/√3)

∴ h√3 = 30 - h/√3

From equation (1)

= (h√3 + h/√3) = 30

= 4h = 30√3 

= h = 15/2 √3m

2. From an aeroplane above the straight road, the angles of depression of two positions at a distance 20 m apart on the road are observed to be 30° and 45°. The height of the aeroplane above the ground is?

Ans: Let the height of the aeroplane above the ground is h and QB = x m.

Given that, PQ = 20m, ∠APB = 30°

and ∠AQB = 45°

Now, in △AQB,

tan 45° = AB/QB = h/x 

⇒ 1 = h/x ⇒ x = h …(i)

and in△ABP, tan30° = AB/PB = h/PQ + QB

⇒ 1/√3 = h/20 + x

⇒ √3b = 20 + x = 20 + b 

From equation(i)

⇒ (√3h - h) = 20

⇒ h (√3 - 1) = 20

∴ h = 20/√3 - 1 * √3 + 1/√3 + 1 

After rationalisation,

= 20(√3 + 1)/(3 - 1) = 20(√3 + 1)/2

= 10(√3 + 1)

Hence, the required height is 

10(√3 + 1)m.