How to Find the Value of Tan 30 Degrees?

Welcome to tan 30°, our post aboutthe tangent of 30 degrees.

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Tan 30 Degrees
Alternative form
Trigonometric functions
Find the value of tan 30°.
Sample Problems
What is the Value of Tan 30 Degrees?
Methods to Find Value of Tan 30 Degrees
Tan 30° in Terms of Trigonometric Functions
Tan 30 Degrees Using Unit Circle
How to Find Tan 30° in Terms of Other Trigonometric Functions?
How to Find the Value of Tan 30 Degrees?
What is the Value of Tan 30° in Terms of Sec 30°?
What is the Value of Tan 30 Degrees in Terms of Sin 30°?

For the tangent of 30 degrees we use the abbreviation tan for the trigonometric function together with the degree symbol °, and write it as tan 30°.

If you have been looking for what is tan 30°, or if you have been wondering about tan 30 degrees in radians, then you are right here, too.

In this post you can find the tan 30° value, along with identities.

Read on to learn all about the tan of 30°.

Tan 30 Degrees

If you want to know what is tan 30 degrees in terms of trigonometry, then navigate straight to the explanations in the next paragraph; what’s ahead in this section is the value of tan 30°:

tan30° = √(3)/3tan 30° = √(3)/3

tan 30 degrees = √(3)/3

The tan of 30 degrees is √(3)/3, the same as tan of 30 degrees in radians. To obtain 30 degrees in radian multiply 30° by $\pi$ / 180° = 1/6 $\pi$. Tan 30degrees = tan (1/6 × $\pi)$.

Our results of tan30° have been rounded to five decimal places. If you want tangent 30° with higher accuracy, then use the calculator below; our tool displays ten decimal places.

To calculate tan 30 degrees insert the angle 30 in the field labelled °, but if you want to calculate tan 30 in radians, then you have to press the swap unit button first.A Really Cool Tangent Calculator and Useful Information! Please ReTweet. Click To TweetBesides tan30°, similar trigonometric calculations on our site include, but are not limited, to:

The identities of tangent 30° are as follows:

tan30°
= cot (90°-30°) = cot 60°

-tan30°= tan (-30°) = -tan 30°= cot (90°+30°) = cot 120°

= tan (180°-30°) = tan 150°Note that tan30° is periodic: tan (30° + n × 180°) = tan 30 degrees, n$\hspace{5px} \in \hspace{5px} \mathbb{Z}$.

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The exact value of tan function if angle of right triangle is $30$ degrees is called tan of $30$ degrees. It’s written as $\tan{(30^°)}$ in mathematical form according to Sexagesimal system.

$\tan{(30^°)} \,=\, \dfrac{1}{\sqrt{3}}$

The exact value of tan of angle $30$ degrees is $\dfrac{1}{\sqrt{3}}$ in fraction from. It is an irrational number and is equal to $0.5773502691\ldots$ in decimal form.

Alternative form

$\tan{(30^°)}$ is written as $\tan{\Big(\dfrac{\pi}{6}\Big)}$ in circular system and also written as $\tan{\Bigg(33\dfrac{1}{3}^g\Bigg)}$ in centesimal system alternatively.

$(1) \,\,\,$ $\tan{\Big(\dfrac{\pi}{6}\Big)} \,=\, \dfrac{1}{\sqrt{3}}$

$(2) \,\,\,$ $\tan{\Bigg(33\dfrac{1}{3}^g\Bigg)} \,=\, \dfrac{1}{\sqrt{3}}$

Proof

You learned the $\tan{\Big(\dfrac{\pi}{6}\Big)}$ value and it’s your time to learn how to $\tan{(30^°)}$ value is derived in trigonometry.

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Trigonometry is the branch of mathematics that deals with the relationship of sides with angles in a triangle. With trigonometry It is easy to find the heights of big mountains or towers, also in astronomy, it is used to find the distance between stars or planets and is widely used in physics, architecture, and GPS navigation systems.

Trigonometry is based on the principle that “If two triangles have the same set of angles then their sides are in the same ratio”. Side length can be different but side ratios are the same.

Trigonometric functions

Trigonometric functions are also called circular functions or trigonometric ratios. The relationship of angles and sides is represented by these trigonometric functions. There are six trigonometric functions Sine, Cosine, Tangent, Cosecant, Secant, Cotangent.

Sin = Perpendicular/ Hypotenuse
Cos = Base/Hypotenuse
Tan = Perpendicular/Base
Cot = Base/Perpendicular
Sec = Hypotenuse/Base
Cosec = Hypotenuse/Perpendicular

Perpendicular, base, and hypotenuse are the three sides of the right-angled triangle. Let’s learn about these terms in detail,

Perpendicular – The side in front of the angle is perpendicular. In this case, the side in front of 30° is called its perpendicular.
Base – A base is one of the sides which touches angle, but hypotenuse can never be considered as base.
What is Hypotenuse – A side opposite to the right angle (90°) is called the hypotenuse, it is the longest side.

Note: Perpendicular and base are not fixed for a triangle. In a triangle, a side is perpendicular for an angle, but the same side is a base for another angle.

As shown in the above diagram for the same triangle if, considered angle 30°, perpendicular is the side PQ, but if we consider angle 60° our perpendicular is side QR.

Find the value of tan 30°.

Solution:

To calculate tan 30°, take ratios of its perpendicular and its base. For which calculate the length of perpendicular and base. To find lengths of sides we take the help of an equilateral triangle.

Take an equilateral triangle of side length 2m.

Now from any vertex draw an altitude.

The altitude drawn divides the equilateral triangle into two right triangles.

Now we have a length of two sides in the right triangle.

The third side is calculated by baudhayan theorem or Pythagoras theorem.

Altitude in equilateral triangle bisect the angle and the side, So angle DAC is 30° and side DC is 1m.

Let the length of side AD is x.

Applying Pythagoras theorem in triangle ADC.

P2 + B2 = H2

AD2 + DC2 = AC2

x2 + 12 = 22

x2 + 1 = 4

x2 = 3

x = √3

So, tan 30° = DC/AD

Tan30° = 1/√3

Once calculated on all sides one can find other trigonometric ratios also, like:

Sin 30° = 1/2
Cos 30° = √3/2

Sample Problems

Question 1: In a right angle triangle, one angle is 30°, and the base for 30° is 3m. Find the length of perpendicular.

Solution:

Given: Base = 3m

Tan 30 = 1/√3

P/B = 1/√3

P/3 = 1/√3

p = √3

Question 2: In a right triangle hypotenuse is 20cm, and one side is 10√3cm, find the angles of the triangle.

Solution:

Given: H = 20, and B = 10√3

Finding third side using pythagoras theorem.

P2 + B2 = H2

P2 + (10√3)2 = 202

p2 + 300 = 400

P2 = 100

p = 10

The third side is 10cm. The ratio of the two sides 10cm and 10√3cm is 1/√3, So there must be an angle of 30° in triangle Since the triangle is a right angle, so the third angle is

90° – 30° = 60°

The Angles of a triangle are 30°, 60°, 90°.

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The value of tan 30 degrees is 0.5773502. . .. Tan 30 degrees in radians is written as tan (30° × π/180°), i.e., tan (π/6) or tan (0.523598. . .). In this article, we will discuss the methods 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: tan (π/6) or tan (0.5235987 . . .)

What is the Value of Tan 30 Degrees?

The value of tan 30 degrees in decimal is 0.577350269. . .. Tan 30 degrees can also be expressed using the equivalent of the given angle (30 degrees) in radians (0.52359 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°) ⇒ 30 degrees = 30° × (π/180°) rad = π/6 or 0.5235 . . .

∴ tan 30° = tan(0.5235) = 1/√3 or 0.5773502. . .

Explanation:

For tan 30 degrees, the angle 30° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 30° value = 1/√3 or 0.5773502. . .
Since the tangent function is a periodic function, we can represent tan 30° as, tan 30 degrees = tan(30° + n × 180°), n ∈ Z. ⇒ tan 30° = tan 210° = tan 390°, and so on.

Note: Since, tangent is an odd function, the value of tan(-30°) = -tan(30°).

Methods to Find Value of Tan 30 Degrees

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

Using Trigonometric Functions
Using Unit Circle

Tan 30° in Terms of Trigonometric Functions

Using trigonometry formulas, we can 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: Since 30° lies in the 1st Quadrant, the final value of tan 30° will be positive.

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°

Tan 30 Degrees Using Unit Circle

To find the value of tan 30 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 30° angle with the positive x-axis.
The tan of 30 degrees equals the y-coordinate(0.5) divided by x-coordinate(0.866) of the point of intersection (0.866, 0.5) of unit circle and r.

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

☛ Also Check:

Tan 30 degrees is the value of tangent trigonometric function for an angle equal to 30 degrees. The value of tan 30° is 1/√3 or 0.5774 (approx).

How to Find Tan 30° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 30° can be given in terms of other trigonometric functions 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°

☛ Also check: trigonometry table

How to Find the Value of Tan 30 Degrees?

The value of tan 30 degrees can be calculated by constructing an angle of 30° with the x-axis, and then finding the coordinates of the corresponding point (0.866, 0.5) on the unit circle. The value of tan 30° is equal to the y-coordinate(0.5) divided by the x-coordinate (0.866). ∴ tan 30° = 1/√3 or 0.5774

What is the Value of Tan 30° in Terms of Sec 30°?

We can represent the tangent function in terms of the secant function using trig identities, tan 30° can be written as √(sec²(30°) - 1). Here, the value of sec 30° is equal to 1.1547.

What is the Value of Tan 30 Degrees in Terms of Sin 30°?

Using trigonometric identities, we can write tan 30° in terms of sin 30° as, tan(30°) = sin 30°/√(1 - sin²(30°)) . Here, the value of sin 30° is equal to 1/2.