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 a right-angled triangle. It is used to solve angles and sides in a right-angled triangle. Sin, cos, and tan are the ratios which 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.

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.

Also, the values of the sin of 30° and cos of 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.

Like Sine and Cosine, Tangent is additionally a basic function of trigonometry. Most of the trigonometric equation is based on these ratios. Usually, to find the values of sine, cosine, and tangent ratios, we use right-angles triangles and also take a unit circle example. First, allow us to discuss tan 30 degrees 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

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

∴ Tan α = sin α/cos α

So, tan 30 degrees we can write 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

For a unit circle also we will calculate the value of tan 30 degrees. The unit circle features a radius as 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.

Question 1. Two poles of equal heights are standing to each other on either side of a road, which is 30 m wide. From 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

Question 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

[rationalisation]

= 20(√3 + 1)/(3 - 1) = 20(√3 + 1)/2

= 10(√3 + 1)

Hence, the required height is

10(√3 + 1)m.

FAQ (Frequently Asked Questions)

1. What is the value of tan 30°?

If an angle of a right-angled triangle is 30° degree, then the value of tan 30°, can be written as tan (30°) according to the Sexagesimal System. If fractional form tan 30°values 1/√3, which is equal to 0.5773502691. It is an irrational number.

2. Why is tan π/2 undefined?

As Tan (x) is the ratio of sin(x) and cos(x). Let, x = π/2, such that

sin π/2 = 1 and cos π/2 = 0 According to the sin, cos and tan relation

sin/cos = tan, Then

1/0 = undefined.

Hence Proved.

3. At what angles tangent is equal to zero?

As we all know, tangent is the ratio of sine and cosine; then, whenever the sin function has a value equal to zero, tan function tends to be zero. Hence, at the angles, 0°, 180°, 270° value of the tangent function is zero.